## 600613

Pick a number, N, then try searching for it on the web via Bing or Google (or maybe the leet version of Google). What can you expect to learn? I wasn’t quite sure of the answer, so I ran some experiments.

When N is a small positive integer—less than 100, say—the leading results tend to be mass-audience web pages that happen to display the numeral N in some prominent way, such as in a headline or a title. There are news stories (Packers 43, Falcons 37), TV stations (WXMI Fox 17), a few brand names (Motel 6), references to iconic events (9/11, Apollo 13), listings of Bible verses (Romans 3:23).

With somewhat larger integers—three or four digits—I see a lot of street addresses, area codes, tax forms, statutes and ordinances. With five-digit numbers, Zip codes become prominent. At six digits we enter the land of hex colors, accompanied by a baffling variety of part numbers, account numbers, serial numbers, patent numbers, error numbers, lottery numbers. With a search string of 8 to 10 digits, telephone directories dominate the results. Still further out on the number line, you eventually come to a numerical desert where Google and Bing usually come up empty.

To get a more quantitative sense of how numbers are distributed across the web, I decided to do some sampling. I randomly selected 2,000 positive integers of 1 to 12 decimal digits, and submitted them to Google as web search queries. To construct the query integers I started with 2,000 floating-point numbers selected uniformly at random (with replacement) from the range $$0 \le m \lt 12$$. For each $$m$$ I calculated $$N = \lfloor 10^{m}\rfloor$$, then ran a Google search for N. The work was done by a Python script with a politeness pause of one second between queries.From the results of each search I extracted $$H(N)$$, the number of hits, which Google reports near the top of the page. Here’s what I found, plotted on log-log scales:

What an intriguing graph! Over most of the range in the log-log plot, the broad trend looks very nearly linear. What does that mean? If the Google data accurately reflect the state of the web, and if my sampling of the data can be trusted, it means the number of web pages mentioning numbers of magnitude $$10^k$$ is roughly constant for all k in the range from $$k = 2$$ to $$k = 10$$. I don’t mean to suggest that specific large numbers appear just as frequently as specific small numbers. That’s obviously untrue: A typical two- or three-digit number might be found on a billion web pages, whereas a specific nine- or ten-digit number is likely to appear on only one or two pages. But there are only 90 two-digit numbers, compared with 90 billion 10-digit numbers, so the overall number of pages in those two classes is approximately the same.

Here’s another way of saying the same thing: The product of $$N$$ and $$H(N)$$ is nearly constant, with a geometric mean of roughly $$7 \times 10^{10}$$. An equivalent statement is that:

$\log_{10}{N} + \log_{10}{H(N)} \approx 10.86.$

You can visualize this fact without doing any arithmetic at all. Just print a series of $$N, H(N)$$ tuples in a column and observe that the total number of digits in a tuple is seldom less than 11 or greater than 13.

    N,        H(N)
96964835, 2120
2048, 164000000
476899618, 214
96416, 374000
75555964, 3020
171494, 182000
154045436, 2160
1206, 112000000
761088, 50200
7500301034, 24
13211445, 10900
1289, 77000000
1507549, 18100
3488, 3330000
7507624475, 10
17592745, 2830
1430187656, 30
691, 265000000
41670244642, 2
326, 52900000

Although the vast majority of the 2,000 data points lie near the 10.86 “main sequence” line, there are some outliers. One notable example is 25898913. Most numbers of this magnitude garner a few thousand hits on Google, but 25898913 gets 29,500,000. What could possibly make that particular sequence of digits 10,000 times more popular than most of its neighbors? Apparently it’s not just an isolated freak. About half the integers between 25898900 and 25898999 score well below 10,000 hits, and the other half score above 20 million. I can’t discern any trait that distringuishes the two classes of numbers. Sampling from other nearby ranges suggests that such anomalies are rare.

A straight line on a log-log plot often signals a power-law distribution. The classic example is the Zipfian distribution of word frequencies in natural-language text, where the kth most common word can be expected to appear with frequency proportional to $$k^{-\alpha}$$, with $$\alpha \approx 1$$. Does a similar rule hold for integers on the web? Maybe. I tried fitting a power law to the data with the powerlaw Python package from Jeff Alstott et al. The calculated value of $$\alpha$$ was about 1.17, which seems plausible enough, but other diagnostic indicators were not so clear. Identifying power laws in empirical data is notoriously tricky, and I don’t have much confidence in my ability to get it right, even with the help of a slick code library.

I’m actually surprised that the pattern in the graph above looks so Zipfian, because the data being plotted don’t really represent the frequencies of the numbers. Google’s hit count $$H(N)$$ is an approximation to the number of web pages on which $$N$$ appears, not the number of times that $$N$$ appears on the web. Those two figures can be expected to differ because a page that mentions $$N$$ once may well mention it more than once. For example, a page about the movie 42 has eight occurrences of 42, and a page about the movie 23 has 13 occurrences of 23. (By the way, what’s up with all these numeric movie titles?)

Another distorting factor is that Google apparently implements some sort of substring matching algorithm for digit strings. If you search for 5551212, the results will include pages that mention 8005551212 and 2125551212, and so on. I’m not sure how far they carry this practice. Does a web page that includes the number 1234 turn up in search results for all nonempty substrings: 1234, 123, 234, 12, 23, 34, 1, 2, 3, 4? That kind of multiple counting would greatly inflate the frequencies of numbers in the Googleverse.

It’s also worth noting that Google does some preprocessing of numeric data both in web pages and in search queries. Commas, hyphens, and parentheses are stripped out (but not periods/decimal points). Thus searches for 5551212, 555-1212, and 5,551,212 all seem to elicit identical results. (Enclosing the search string in quotation marks suppresses this behavior, but I didn’t realize that until late in the writing of this article, so all the results reported here are for unquoted search queries.)

In the graph above, the linear trend seems to extend all the way to the lower righthand corner, but not to the upper lefthand corner. If we take seriously the inferred equation $$N \times H(N) = 7 \times 10^{10}$$, then the number of hits for $$N = 1$$ should obviously be $$7 \times 10^{10}$$. In fact, searches for integers in the range $$1 \le N \le 25$$ generally return far fewer hits. Many of the results are clustered around $$10^{7}$$, four or five orders of magnitude smaller than would be expected from the trend line.

To investigate this discrepancy, I ran another series of Google searches, recording the number of hits for each integer from 0 through 100. Note that in this graph the y axis is logarithmic but the x axis is linear.

There’s no question that something is depressing the abundance of most numbers less than 25. The abruptness of the dip suggests that this is an artifact of an algorithm or policy imposed by the search engine, rather than a property of the underlying distribution. I have a guess about what’s going on. Small numbers may be so common that they are treated as “stop words,” like “a,” “the,” “and,” etc., and ignored in most searches. Perhaps the highest-frequency numbers are counted only when they appear in an <h1> or <h2> heading, not when they’re in ordinary text.

But much remains unexplained. Why do 2, 3, 4, and 5 escape the too-many-to-count filter? Same question for 23. What’s up with 25 and 43, which stand more than 10 times taller than their nearest neighbors? Finally, in this run of 101 Google searches, the hit counts for small integers are mostly clustered around $$10^6$$, whereas the earlier series of 2,000 random searches produced a big clump at $$10^7$$. In that earlier run I also noticed that searching repeatedly for the same $$N$$ could yield different values of $$H(N)$$, even when the queries were submitted in the space of a few seconds. For example, with $$N=1$$ I saw $$H(N)$$ values ranging from 10,400,000 to 1,550,000,000. Presumably, the different values are coming from different servers or different data centers in Google’s distributed database.

I was curious enough about the inconsistencies to run another batch of random searches. In the graph below the 2,000 data points from the first search are light blue and the 2,000 new points are dark blue.

Over most of the range, the two data sets are closely matched, but there’s a conspicuous change in the region between $$10^2$$ and $$10^4$$. In the earlier run, numbers in that size range were split into two populations, with frequencies differing by a factor of 10. I was unable to identify any property that distinguishes members of the two populations; they are not, for example, just odd and even numbers. In the new data, the lower branch of the curve has disappeared. Now there is a sharp discontinuity at $$N = 10^4$$, where typical frequency falls by factor of 10. I have no idea what this is all about, but I strongly suspect it’s something in the Google algorithms, not in the actual distribution of numbers on the web.

The limitations of string matching—or even regular-expression matching—are more troublesome when you go beyond searching for simple positive integers. I’ve hardly begun to explore this issue, but the following table hints at one aspect of the problem.

N top hit
17.3 HP Anodized Silver 17.3″ Pavilion
17.30 17.30j Syllabus – MIT
17.300 Chapter 17.300 COMPLIANCE
17.3000 Map of Latitude: 17.3000, Longitude: -62.7333
17.30000 41 25 0 0 2.000000 4.000000 6.000000 8.000000
17.300000 17.300000 [initially -35.600000] gi_24347982 (+) RNA

Search queries that are mathematically equal (when interpreted as decimal representations of real numbers) yield quite different results. And 4.999… is definitely not equal to 5.000… in the world of Google web search.

It gets even worse with fractions. A search for 7/3 brought me a calculator result correctly announcing that “7/3 = 2.33333333333″ but it also gave me articles headlined “7^3 – Wolfram Alpha”, “Matthew 7:3″, “49ers take 7-3 lead”, and “Hokua 7’3″ LE – Naish”. (Enclosing the search term in quotation marks doesn’t help in this case.)

Before closing the book on this strange numerical diversion that has entertained me for the past couple of weeks, I want to comment on one more curious discovery. If you run enough searches for large numbers, you’ll eventually stumble on web sites such as numberworld, numberempire, numbersbase, each-number, every-number, all-numbers, integernumber, numbersaplenty, and numberopedia. A few of these sites appear to be created and curated by genuine number-lore enthusiasts, but others have a whiff of sleazy search-engine baiting. (For that reason I’m not linking to any of them.)

Here’s part of a screen capture from Numbers Aplenty, which is one of the more interesting sites:

Each of the numbers displayed on the page is a link to another Numbers Aplenty page, and the site is apparently equipped to display such a page for any positive integer less than $$10^{16}$$. A few years ago, Google reported that they had indexed a trillion unique URLs on the world wide web. Evidently they hadn’t yet worked their way through the 10,000 trillion URLS at Numbers Aplenty. (But I’m pretty sure the server doesn’t have $$10^{16}$$ HTML files stored on disk, patiently waiting for someone to request the information.

And, finally, a trivia question: What is the smallest positive integer for which a Google search returns zero results? The smallest I have recorded so far is 10,041,295,923. (Of course that could change after the Googlebot indexes this page.) Can anyone find an example with 10 or fewer digits?

Update 2014-12-22. Commenter Samuel Bierwagen wrote:

The hits number on the first page is very inaccurate, frequently off by several orders of magnitude. To get better results you have to go to the second or third page.

I’ve now given this idea a try. Whenever the estimated hit count is at least 1 million, I repeat the search, appending “start=20″ to the query string. This has the effect of requesting the third page of results (i.e., results 20 through 29). Here’s the outcome:

Light blue dots are from earlier surveys (4,000 points in all). Dark blue dots are from the new survey, with the page-three request installed. There’s a dramatic change in the hit counts $$H(N)$$ for $$N \lt 100$$. For these small $$N$$, Google’s first-page hit count fluctuates wildly and is often near $$10^{7}$$. The third-page results are higher and much more consistent. Indeed, all $$N \lt 14$$ returned exactly the same hit count: 25,270,000,000. This uniformity suggests that we’re still seeing some sort of filtering in the results–I suspect Google may be trying to keep secret the overall size of their index–but at least the trend line is now monotonic.

In another comment, Brian J. Peterson mentions seeing HTTP results with an error code 503 (service temporarily unavailable). I had not encountered any such errors in my earlier search series, but I did see some in this latest run (86 errors out of 2,000 searches). My best guess is that a request for page 3 may occasionally take more than 1 second, so that the transaction hasn’t completed when the next search is initiated.

Meanwhile, I have learned of another study of number prevalence on the web with a much better data source than Google hit counts. In a short paper from the 2014 World Wide Web companion conference, Willem Robert van Hage and two colleagues used the Common Crawl web archive to measure number frequencies. They looked at real numbers, not just integers, and I’m not sure how to compare their results with mine. My main response to seeing this work is that the Common Crawl is an amazing resource–each of us can build a Google of our own–and I want to spend some time next year playing with it.

Posted in computing, mathematics | 20 Comments

## Four Fifths = A Fifth

Browsing in the library stacks the other day, I came upon this passage in a 1970 paper discussing the uses of computers in mathematics:

Notice anything fishy about that equation? On the right side of the equal sign we have an odd number of odd numbers, and so their sum must be odd; but the power of 144 on the left side is even.

I’m not clever enough to have caught that parity violation on first glance; I noticed it only later. On the other hand, my experience as an editor has taught me never to trust an author’s arithmetic. I had a computer sitting in front of me, open to an IPython notebook, so I typed in the numbers:

Had I just discovered that a 50-year-old counterexample to a 250-year-old conjecture is not in fact a counterexample after all? Surely not. Someone must have erred in copying the equation from the original source.

So I decided to track down that source. The reference in the quoted passage directed me to a 1966 paper by L. J. Lander and T. R. Parkin, which I quickly found online. However, the reference is mistaken; the cited paper says nothing at all about fifth powers and has no equation resembling the one above.

A faulty equation and a wayward reference within a single paragraph: That’s not so good. But I’ve done worse myself, and my purpose here is not to criticize an errant author (whom I’m not going to name). What interests me is how one might go about correcting the error in the equation and finding the true counterexample to the Euler conjecture. Here are some strategies I might have tried that morning:

• I was sitting in a university library. I had all the apparatus of scholarly research at my fingertips. I could turn to MathSciNet, say, and identify the correct paper by Lander and Parkin.
• I was sitting at a computer with a wifi connection. I had all the resources of the Internet at my fingertips. I could Google it, or consult some more specialized database such as the OEIS.
• I could just guess the answer. On the assumption that exactly one of the five terms has been altered, I would expect to come up with the correct equation after just a few tries.
• I could start from scratch. I could write a program to repeat the computer search that Lander and Parkin evidently used to discover the equation in the first place.
• I could bring the might of mathematics to bear on the problem. That is, I could try to “solve the equation.”

At this point you might want to stop reading, choose your own favorite method, and give it a try. The answer, once you have it, is not all that illuminating, but I found the process of searching had its moments of interest.

If guessing is one of your strategies, you had better try it first. I already had the equation entered into the IPython notebook, where I could easily alter a value and observe the result. (A spreadsheet would serve as well in this role.) I started with the $$27^5$$ term, reducing it to $$26^5$$, then $$25^5$$, and so on, but the effect was too small. Even setting that term to zero or to $$(-27)^5$$ left an excess, so I moved on to the $$85^5$$ term. Proceeding in this way, it didn’t take long to pinpoint the error.

The reference is: L. J. Lander and T. R. Parkin, 1966, Counterexample to Euler’s conjecture on sums of like powers, Bulletin of the American Mathematical Society 72:1079. The text of this paper consists of five lines, one of which is the equation.Googling also produced useful results, including pages on Wikipedia and MathWorld that give the correct equation and the correct reference. But finding those pages was not straightforward, and the experience has left me wondering how best to search for numerical or mathematical content on the web. You can always go after the metadata; in this case the names Lander, Parkin, and Euler are good choices, as well as the terms “nth power” and “conjecture.” You might also get lucky searching for character strings such as “144^5” and “27^5”, but here the outcome is hit or miss. There are many ways of encoding mathematics in digital documents—TeX, MathML, the ASCII pidgin that evolved years ago on Usenet, notations based on programming languages. Without some higher-level semantic processing, no single search string will find them all. Some web sites (including MathWorld) turn equations into tiny GIF images, making search even more difficult.

A question I haven’t been able to answer through web search is whether the erroneous equation from that 1970 paper has propagated into the subsequent literature. I have found no evidence of it, but I have little confidence in the ability of current search technology to answer the question reliably. Gathering quantitative data—how many times does the equation appear on the web?—also seems to be out of reach.

The issue goes beyond Google and the other web search engines. MathSciNet claims to accept search strings in TeX format, but I’ve never figured out how to make it work. As for the OEIS, the search function there is specialized for finding integer sequences—as one might expect in an Online Encyclopedia of Integer Sequences. Searching for an equation is an “off label” use of the archive. Still, I was sure I’d find it. I tried searching for 61917364224 (which is $$144^5$$), and got 47 hits. (It’s a popular number because it’s a product of many small factors, $$2^{20} 3^{10}$$, and is also a Fibonacci number.) However, none of those 47 sequences has anything to do with the Euler sum-of-powers conjecture or the Lander-Parkin counterexample. Yet the equation I was looking for does indeed appear in the OEIS, namely in sequence A134341.

The next approach—redoing the entire computation from scratch—turned out to be easier than I expected. Lander and Parkin ran their search on a CDC 6600, which was the most powerful computer of its time, the world’s first megaFLOPS machine, designed by Seymour Cray early in his career. I don’t know how long the computation took on the 6600, but there’s not much to it on a modern laptop. Here’s some Python code:

import itertools as it

def four_fifths(n):
"""Return smallest positive integers ((a,b,c,d),e) such that
a^5 + b^5 + c^5 + d^5 = e^5; if no such tuple exists
with e < n, return the string 'Failed'."""
fifths = [x**5 for x in range(n)]
combos = it.combinations_with_replacement(range(1,n), 4)
while True:
try:
cc = combos.next()
cc_sum = sum([fifths[i] for i in cc])
if cc_sum in fifths:
return(cc, fifths.index(cc_sum))
except StopIteration:
return('Failed')

The first step here is to precalculate the fifth powers of all integers less than a given limit, n. Then we generate all combinations of four integers in the range from 1 to n, starting with the 4-tuple (1, 1, 1, 1) and continuing to (n, n, n, n). For each combination we look up the fifth powers of the integers and check to see whether their sum is also present in the precomputed vector of fifth powers. Invoking this program as four_fifths(150) returns the correct answer in about 30 seconds. A little sad, isn’t it? This was once a research project fit for a supercomputer, and now it’s reduced to the level of a homework assignment.

The final tool we might apply to this problem is the hammer of mathematics. I would put the question as follows. We are looking for integer solutions to this equation:

$a^5 + b^5 + c^5 + d^5 - e^5 = 0.$

In that quest, do we gain any significant leverage by knowing that:

$27^5 + 85^5 + 110^5 + 133^5 - 144^5 = 254933701 ?$

Can we take the number 254933701 and apply some algebraic or analytic hocus pocus that will lead us directly to the correct values of a, b, c, d, e? If we make no further assumptions or guesses, I believe the answer must be no. After all, there are innumerable values for a, b, c, d, e that we can plug into our equation to get something other than 0. Consider this example:

$30^5 + 69^5 + 82^5 + 86^5 - 100^5 = 43.$

That’s a very near miss—it comes within 5 parts per billion of being a true solution—and yet it tells us nothing about the set of correct solutions except that this isn’t one of them.

An erroneous equation seems to be useful only if we know quite a lot about the nature of the errors. For example, in the faulty equation from 1970, if someone tips you off that all the terms are correct except for $$85^5$$, you can form the equation

$85^5 - x^5 = 254933701$

and solve for $$x$$. If $$x$$ turns out to be an integer, you have your answer. And indeed this is the case:

$x = \sqrt[5]{4182119424} = 84.$

But is this procedure better than guesswork? Is it different from guesswork?

Just for the record, the correct Lander-Parkin equation is:

$27^5 + 84^5 + 110^5 + 133^5 = 144^5 = 61917364224.$

After discovering their counterexample to the Euler conjecture, Lander and Parkin, together with John Selfridge, made some conjectures of their own. Suppose an nth power can be partitioned into k smaller nth powers. They conjectured that k is never less than $$n-1$$. And they asked whether there always exists an example where $$k = n-1$$. You might try to settle the latter question for the case of $$n = 6$$. Can you find a set of five 6th powers that add up to another 6th power? For this problem, Google won’t help you, because the answer is unknown. No one has even identified a set of six 6th powers that sum to a 6th power. If you approach this task as a computational challenge, it’s rather more than a homework assignment.

Posted in computing, mathematics | 8 Comments

## Lopsided

Sifting through the election results last week, I noticed that the precinct where I used to live in Durham, North Carolina, voted 620 to 40 in favor of the Blue candidate in the U.S. Senate race. That’s a margin of just under 94 percent. A few nearby polling places were even more lopsided: One score was 288 to 7, which works out to 97.6 percent. Statewide, however, the contest was quite close; the Blues lost with 49.13 percent to the Reds’ 50.87 percent. (The percentages I’m giving here are based on votes for the two major parties only; published results are slightly different because they include votes for third-party candidates and write-ins.)

The combination of one-sided local results and a nearly even split in the statewide totals left me curious about the distribution of vote margins, or what we might call the political polarization spectrum. Are extreme ratios, like those in my old neighborhood, rare outliers? Or have we become a nation of segregated political communities, where you live with your own kind and stay away from places where the other side dominates?

The infographics group at The New York Times has produced a series of election maps that offer much geographic insight.

Portion of a New York Times map showing voting patterns in the 2014 U.S. Senate election in North Carolina. Durham is the blue inkblot near the right edge. Graphics by Amanda Cox, Mike Bostock, Derek Watkins, and Shan Carter.

The North Carolina map shows deep blue city cores surrounded by ruddy or purplish suburbs. Rural areas, with lower population density, range from pink to baby blue. The maps are lovely—they transform ugly politics into luscious cartography—but they can’t give a quantitative answer to my question about political segregation and integration.

So off I went to the web site of the North Carolina Board of Elections, where I pulled down the latest report (still unofficial), giving vote totals for the state’s 2,726 precincts. After some vigorous data wrangling (Excel, Textmate, Python, Lisp, eyeballing), I had a histogram with 20 bins, classifying precincts according to their percentage of votes for the Red candidate.

Before I show the data, it’s worth pausing to think about what the distribution might be expected to look like. Suppose the state’s population were thoroughly mixed, with Red and Blue voters scattered at random. Then the precinct voting margins would follow a normal distribution, with most of the precincts near the 50-50 mark and only a few out in the far-right and far-left tails. The curve might look something like this:

Actually, that’s not quite what the curve would look like; it would be much narrower. Suppose you take 3,000 precincts with 1,000 voters each, and you paint the individual voters Red or Blue at random with probability 1/2. You would get a normal distribution with a mean of 500 and a standard deviation of about 16 voters (i.e., $$1/2\sqrt{1000}$$). Almost all of the precincts would lie within three standard deviations of the mean, or in other words between 450 and 550 Red voters. The histogram would be more like this:

Of course nobody who reads the newspaper would suggest that random mixing provides a good model of the American political landscape. Given the evidence of increasing polarization in the public, we might expect to find a two-humped camel, with a Red peak and a Blue peak, and not much of anybody in the middle:

Or maybe the distribution is even more extreme: an empty bowl, with every precinct drifting centrifugally toward the outer fringes of the political spectrum:

All right, I’ve stalled long enough; no more playing with toy distributions. Here’s the real deal—the histogram based on North Carolina precinct data for the recent Senate election:

The shape of this curve took me by surprise, mostly because of its strong asymmetry. Remember, the total numbers of Red and Blue voters in the state are roughly equal, but the apportionment of Reds and Blues in precincts turns out to be quite different. At the left edge of the graph there are 150 precincts where at least 90 percent of the voters chose the Blue candidate, but over on the right side there are only 8 precincts that voted at least 90 percent Red.

Please don’t misinterpret this graph. It doesn’t say that Blue voters are more extreme or more radicalized than Red ones. Nothing of the sort. It says that Blue voters tend to huddle together in more homogeneous communities. If you’re Blue and you want to be surrounded by like-minded voters, there are hundreds of places you can live. If you’re Red, you have lots of choices where you’ll be in the majority—more than 60 percent of the precincts satisfy that criterion—but you would have a hard time finding areas where you have few or no Blue neighbors.

The pattern seemed peculiar enough that I began to wonder if it might be unique to North Carolina. So I took a look at the Senate race in Virginia, which was even closer than the North Carolina contest. The Virginia pattern differs only in detail. Again we have an abundance of solidly Blue precincts, but very few solidly Red ones:

Could it be something about the South that produces this skew in the curve? I wanted to check the closely contested governor’s race in Massachusetts (where I live now), but the state hasn’t yet posted precinct-level results. Instead I looked at the Minnesota Senatorial election:

Here the raised shoulder on the Blue side is not nearly as dramatic, but the asymmetry is present. (Note that this race did not have a close finish; the Blues won by 10 percentage points.)

There are several ways we might try to explain these patterns—or explain them away. A glance at the Times maps, with their inky blue urban neighborhoods, suggests it’s all about city living. But this impression is partly an artifact of the maps’ graphic scheme, which uses color to encode not just voting margin but also population density. As a result, strongly skewed votes in rural areas are not nearly as conspicuous as those in cities. In North Carolina there are several pale blue precincts in the countryside that have vote totals above 85 percent Blue. Minnesota exhibits the same pattern; indeed, one Iron Range precinct, in the far north of the state, recorded a clean sweep for the Blues, 218 to 0. In Virginia, on the other hand, all of the most strongly biased precincts seem to be urban. And it remains true that the vast majority of voters in lopsided precincts are city dwellers.

It’s also important to note that city districts are geographically much smaller than rural or suburban ones. Perhaps this fact alone could explain much of the Blue-Red difference. If we carved up the rural districts into areas the same size as the city precincts, would we find homogeneous clusters of voters? I have no data to answer that question one way or the other. But even if such clusters do exist, the situation remains asymmetric because the Blue voting blocks are larger in population by an order of magnitude.

Race is another factor that can’t be ignored. Those strongly Blue precincts in rural North Carolina are also largely black precincts. But race is not the whole story. Many of the Bluest urban districts are racially and ethnically diverse.

In the end, what’s most puzzling about the histograms is not just the existence of many pure Blue precincts but the near absence of pure Red ones. What accounts for this imbalance? It’s not hard to imagine social mechanisms that would separate people into affinity groups, but the simplest models are symmetrical. We need to find a factor that acts differently on the two parties.

I have not settled in my own mind what I think is going on here, but I would like to offer three possible mechanisms for consideration.

First, maybe Red voters and Blue voters differ in the criteria they apply when choosing a place to live. We might test this idea in a variant of the Schelling segregation model, in which people tend to go elsewhere when they have too few neighbors of their own kind. Perhaps Red voters refuse to live where the proportion of Reds is less than 1/3, but Blues are content to stay even where they are a tiny minority. Alternatively, we might suppose that when Blues reach a 2/3 majority, they drive out the remaining Reds, whereas Reds are willing to tolerate a Blue minority in their midst. In either case, the result is “Red flight” from strongly Blue areas but no countercurrent of Blue voters fleeing Red areas. I think this model might be capable of explaining the observations, but I have no idea how to explain the model. Is there any evidence for such an asymmetry in personal preferences and behavior?

The second possibility is that Blues are more effective than Reds in persuading their neighbors to support the local political majority. In other words, it’s not that Reds flee or are expelled from Blue areas; rather, they are converted into Blues. Then we have to ask: How come the Reds are unable to win over their own Blue neighbors?

My third candidate explanation is gerrymandering. Maybe what we’re seeing is not some natural tendency to form uniformly Blue communities but rather an attempt to draw the boundaries of precincts in a way that concentrates Blue voters in certain districts, leaving the rest of the precincts with a Red majority. To test this hypothesis we might compare states in which different parties dominate the state legislature and thus control the redistricting process. If the asymmetry really is caused by an attempt to hem in the minority party, we should see a mirror image in states where the other party is in power. The three states I’ve looked at so far are not a useful sample in this respect: North Carolina and Virginia both have Red legislatures, and Minnesota’s was also controlled by the Reds during the most recent redistricting cycle. Massachusetts will be a good test case when the numbers come out.

One last thought: In this essay I have written about Reds and Blues rather than Republicans and Democrats in an attempt to keep the focus on a mathematical question and to keep my distance from partisan passions. For the record, however, I don’t actually believe that politics is a game played by brightly colored teams. And I do take sides. Last Tuesday, in my opinion, the stinkers won.

Posted in social science, statistics | 19 Comments

## Revised and Updated

The new edition of my big picturebook, Infrastructure: A Guide to the Industrial Landscape, was published October 27th and is now on display in better bookstores everywhere. Above is the cover; below is the book as seen in the wild—on the shelves of the Harvard Coop and the MIT Press Bookstore. And, yes, you can also get it at Amazon.

I am late in posting this announcement because I wanted to redo the book’s web site, and that took a week longer than expected. (I’m still not entirely happy with the design and the user interface, but that’s a story for another time.)

So what’s revised and updated in the revised and updated edition? It’s been almost a decade since the first edition came out, which is ages in Internet years. And, as a matter of fact, the chapter on communication technology was the one most in need of updating. No one seems much interested in fax machines and beepers these days; brushing those topics aside made room for a new section on data centers. The petroleum industry has also changed dramatically: Fracking is no longer a novelty, and worries about “peak oil” seem to have been displaced by “the new Saudi America.” Meanwhile aviation is on the cusp of a great change, as the whole network of ground-based navigational beacons is about to be dismantled in favor of GPS.

On the other hand, I couldn’t find much of anything new to say about American railroads. They still represent the pinnacle of development of 19th century technology.

It’s not only the landscape that has changed; I have changed too, and I would now offer a somewhat more cautious assessment of how technology serves human needs. Not that I’m about to join the Luddites; I still believe that if we’re going to accommodate 10 billion people on this planet, we need all the technological help we can muster. But I’m not greatly impressed with our social and economic mechanisms for choosing which technologies to pursue.

The new edition gave me a welcome opportunity to correct some errors, to add a few new photos and to improve the presentation of all the photos. (There are more than 750.) And I’m pleased to report that the publisher (W. W. Norton) and the printer (a branch of R R Donnelley in China) have done quite a spiffy job of bookmaking. They’ve also held the price at \$35, the same as the older paperback.

If you read the first edition, should you run right out to buy a copy of this new one? I won’t try to talk you out of it. Admittedly, this is not a new book; it’s an improved version of the old one. But I’d like to think it’s worth reading again, and you might as well have the benefit of the latest word.

Posted in books | 2 Comments

## Sunshine In = Earthshine Out

No computer simulations have ever had broader consequences for human life than the current generation of climate models. The models tell us that rising levels of atmospheric carbon dioxide and other greenhouse gases can trigger abrupt shifts in the planet’s climate; to avert those changes or mitigate their effects, the entire human population is urged to make fundamental economic and technological adjustments. In particular, we may have to forgo exploiting most of the world’s remaining reserves of fossil fuels. It’s not every day that the output of a computer program leads to a call for seven billion people to change their behavior.

Thus sayeth I in my latest American Scientist column (HTML, PDF). Climate is a subject I take up with a certain foreboding. The public “debate” over global warming has become so shrill and dysfunctional that I can barely force myself to pay attention, much less join the fray. As I worked on writing the column, I found myself tiptoeing through the polemical minefield, carefully avoiding any phrase that might be picked up by “the other side” and used against me. (“Even a writer for American Scientist has doubts about . . .”). When I handed in the text, my editors went over it with the same cautious anxiety—and they found a few places where they thought I hadn’t been careful enough.

This is not the kind of science I enjoy. I would much prefer to dwell in less-contentious corners of the cosmos, where I can play with my sticky spheres or my crinkly curves, and never give a thought to “the other side.” But now and then one must return to the home planet. Besides, the computer modeling that plays a major role in climate science is just my kind of thing.

So, for the past few months climate models have been my breakfast, lunch, and dinner, and my bedtime snack. I’ve been reading in the literature, and poring over source code; I’ve managed to get a couple of serious models running on a laptop. The American Scientist column says more about the rewards and frustrations of all these undertakings. Here I want to talk about a lighter side of the project: a little climate model of my own. The image below is just a static screen grab, but you can go play with the real thing if you’d like.

Okay, it’s not really all my own. The physics behind the model was explored more than 100 years ago by Svante Arrhenius. The first computer implementations were done by M. I. Budyko and William Sellers in 1969, and a third version was published by Stephen Schneider and Tevi Gal-Chen in 1973. But the JavaScript is mine. (Any mistakes are mine, too.)

The model is a simple one. The planet it describes is not the Earth we know and love but a bald sphere without continents or oceans, seasons or storms, or even day and night. Temperature is the only climatic variable. The globe is divided into latitudinal stripes five degrees wide, and temperatures are always uniform throughout a stripe. Other than warming and cooling, the only thing that can happen on this planet is freezing: If the temperature in a stripe falls below –10 degrees Celsius, it grows an ice sheet.

Why pay any attention to such a primitive model? Although a zillion details are missing, some important physical principles emerge with particular clarity. At the heart of the model is the concept of energy balance: If the Earth is to remain in thermal equilibrium with its surroundings, then it must radiate away as much energy as it receives from the Sun. The planet’s temperature will rise or fall in order to maintain this balance. The governing equation is:

$Q (1 - \alpha) = \sigma T^{4}.$

Here $$Q$$ is the average intensity of incoming solar radiation in watts per square meter, $$\alpha$$ is the albedo, or reflectivity, and $$T$$ is temperature in Kelvins; $$\sigma$$ is the Stefan-Boltzmann constant, which relates thermal emission to temperature. The numerical value of $$\sigma$$ is $$5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}}$$. In other words, sunshine in = earthshine out.

If we know the solar input, we can calculate the Earth’s temperature at equilibrium by solving for $$T$$:

$T = \sqrt[4]{\frac{Q (1 – \alpha)}{\sigma}}$

This is exactly what the energy-balance model does.

Two more effects need to be taken into account. First, most of the sun’s energy is received in the tropics, but it doesn’t all stay there. On the Earth, heat is redistributed by the process we call weather—evaporation and condensation, winds, storms, etc.—and also by ocean currents. In the model, the effect of all that swirly fluid dynamics is crudely approximated by a diffusive flow that simply smooths out temperature gradients.

Finally, there’s the greenhouse effect. Solar radiation at visible wavelengths passes through the Earth’s atmosphere to warm the surface, but the Earth emits at longer wavelengths, in the infrared; radiation in that part of the spectrum is absorbed by water vapor, carbon dioxide, and other minor constituents of the atmosphere. The blocked infrared emission raises the temperature at the surface by 30 degrees Celsius or more. The Earth would be a very chilly place without this effect, but the burning issue of the moment is how to avoid overheating. The model sidesteps all the complexities of atmospheric chemistry and absorption spectra; a single parameter, H, simply determines the fraction of outgoing radiation trapped by greenhouse gases.

The interface to the model has four controls—sliders that adjust the incident solar flux, the albedo of land, the albedo of ice-covered regions, and the greenhouse parameter. While playing with the slider settings, it’s not hard to get the model into a state from which there is no easy recovery. That’s what the reset button is for. (Too bad the real planet doesn’t have one.)

Some experiments you might try:

• The default greenhouse setting is H = 0.4, meaning that 40 percent of the outgoing radiation is intercepted. Lower the slider to below 0.2. The model enters a “snowball Earth” state, where ice sheets descend all the way to the Equator. Now return the greenhouse slider to the default setting of 0.4. The ice persists, and it will not melt away until the control is moved to a still higher setting. When the thaw does come, it is sudden and overshoots, leaving the planet in a much warmer state.
• Raise the greenhouse slider to about 0.45, where the polar ice cap disappears. Returning the slider to 0.4 does not restore the ice cap, and the polar areas remain several degrees warmer than they were before the greenhouse excursion.
• Raising the land albedo (so that more sunlight is reflected from snowfree surfaces, and less is absorbed) reveals another “ratchet” mechanism. Once the planet becomes totally icebound, further changes in the land albedo have no effect at all. The reason is simply that no ice-free land is exposed.
• Push the greenhouse slider toward the top of the scale. Beyond H = 0.75, tropical temperatures approach the boiling point of water, and the planet has obviously become uninhabitable. Question: What will happen at H = 1.0, where no radiation at all can escape the atmosphere?

Some of the extreme parameter values mentioned above yield fanciful results that would never be seen on the Earth. But certain interesting aspects of the model seem quite plausible. I am particularly intrigued by the presence of abrupt changes of state that look much like phase transitions. If we consider just the value of the greenhouse parameter H and the average global temperature T, we can draw a two-dimensional phase diagram in the (H, T) plane. The plot below traces the system’s trajectory as H is lowered from 0.4 to 0.2, then raised to 0.5, and finally returned to the initial setting of 0.4.

That four-sided loop is the hallmark of hysteresis (a term first introduced in the study of magnetic materials). Initially, as the greenhouse effect weakens, temperature falls off linearly. Then, at about H = 0.3, the curve steepens. (The stairsteps represent the freezing of successive zones of latitude.) Just below H = 0.218, the temperature falls off a cliff, dropping suddenly by 20 degrees Celsius. On the next segment of the curve, as H increases again, the temperature again responds linearly, but in a much colder regime. Only when H reaches 0.459 is warmth restored to the world, and this time there’s an abrupt upward jump of almost 35 degrees.

It’s no mystery what’s causing this behavior. There’s a strong feedback effect between cooling/freezing and warming/thawing. When the temperature in a latitude band falls below the –10 degree threshold, the zone ices over; the ice-covered surface is more reflective, and so less heat is absorbed, cooling the planet still more. Going in the other direction, when the ice melts, the exposed land absorbs more heat and brings still more warming.

The sharp, discontinuous transitions in the graph above could not happen on the real planet. They are possible in the model because it has no notion of heat capacity or latent heat; after every perturbation, the system instantly snaps back to equilibrium. But if the hysteresis loop in the model has unrealistically sharp corners, its basic shape is not impossible on a physical planet. What’s most important about the loop is that over a wide range of greenhouse parameters, the system has two stable states. At H = 0.4, for example, the world can have an average temperature of either 14 degrees of –22 degrees. That kind of bistability may well be possible in terrestrial climate.

Snowball Earth is not a fate we need to worry about anytime soon, but there is evidence that the planet actually went through such frozen states early in its history. The runaway greenhouse effect that would boil away the oceans is also not an immediate threat. So can we rest easy about living on a sedate, linear segment of the H-T curve? That’s a question that climate models are supposed to answer, but it’s beyond the scope of this particular model.

Posted in computing, physics | 3 Comments

## The pi man

Some weeks ago a reader sent me a note about an old article of mine, written back in 1989. The PDF, he reported, was missing a page. So I went scrounging through some untidy heaps of papers, found a copy, scanned it, and posted a new PDF. But along the way I stopped to reread my own words from 25 years ago.

The piece was a review of the first version of Mathematica, which debuted that summer of 1989. I began my review:

Sometime in the 1850′s William Shanks, a British schoolkeeper and amateur mathematician, began calculating the digits of the decimal expansion of π. Twenty years later, when he published his results in the Proceedings of the Royal Society, he had gotten as far as 707 digits. Today, with the aid of the computer program Mathematica, performing the same calculation requires no more effort than typing the expression:

N[Pi, 707] .

In response to this command, the labor that occupied Shanks for two decades is completed in less than eight seconds. Furthermore, Mathematica produces the correct 707 digits. Shanks made an error in the 528th decimal place that spoiled all the rest of his work.

I remember being wowed by the sub-eight-second running time of the π computation. (My notes say the actual time was 7.97 seconds.) That was with Mathematica 1.0 on a Macintosh SE/30 (16 MHz 32-bit processor, 5 MB of memory). How long would it take with more recent hardware and software? Naturally I had to try the experiment. Wolfram Research has just released version 10.0 of Mathematica. I ran it on the modern equivalent of the SE/30: a middle-of-the-road Mac laptop (2.6 GHz 64-bit processor, 8 GB of memory). The result: 190 microseconds.

I’m wowed again.

We don’t know how long Shanks actually spent noodling away at the digits of π. He did not—despite what I wrote in 1989—work continuously for two full decades. One known data point is that he extended his result from 530 decimal places to 607 places during March and April of 1853. Suppose he gave the task 60 hours a week during those two months. That’s roughly 500 hours for 77 digits, which we might extrapolate to 5,000 hours for 707. The average computation speed is thus 0.15 digits per hour, or 0.000043 digits per second. Compare this performance with later results:

Computer Year Speed
Shanks with a pencil 1873 0.000043
ENIAC with punch cards 1949 0.0079
16 MHz Mathematica 1.0 1989 89
2.6 GHz Mathematica 10.0 2014 3.7 × 106

What I find surprising in these numbers is that the ENIAC was only a few hundred times faster than a Victorian English gentleman wielding pencil and paper, whereas my current laptop is a billion times faster than the ENIAC. Much of this recent speedup is technological—a blessing of Moore’s Law. But algorithmics surely plays a role too.

As I mused about these developments, I realized that I had no idea how Mathematica computes the numerical value of π, or whether the method might have changed between version 1.0 and version 10.0. I also had only a vague notion of how Shanks did his computation. It turns out that both he and the ENIAC programmers used something called Machin’s formula:

$\frac{\pi}{4} = 4\arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)$

Thus they found the value of π by first finding the value of two arctangents. That’s good to know, but it raises a further question: How do you evaluate the arctangent of 1/5 or 1/239? You can’t just look it up in a table if you need 700+ digits of precision. The solution, I learned, is to evaluate some terms of an infinite series, and then sum them up:

$\arctan x = \frac{x^{1}}{1} - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots = \sum_{k=0}^\infty \frac{(-1)^{k}}{2k+1}x^{2k+1}$

Mathematically, this is all you really need to know to understand what Shanks was doing in his computation of π. The whole job reduces to elementary arithmetic: Adding and subtracting, multiplying and dividing. But this was arithmetic at scale. I was having a hard time imagining the logistical and mechanical details of working with hundreds of numbers that run to hundreds of decimal places. “More is different” is a slogan from physics, but surely it applies to arithmetic too.

The last time I mentioned Shanks in these pages, I heard from Erwin Engert, who has done some thoughtful analysis of Shanks’s methods (and his errors). The documents on Engert’s web site include records of his own pencil-and-paper computations of π to 20 and 40 decimal places. I’m not brave enough to try that myself, but I thought I might be able to learn something useful by writing programs that follow some of the same steps a human solver might choose. The project would take a day or two, I supposed. Then one thing led to another (as it has a habit of doing).

For many of us, William Shanks is a figure to be pitied or ridiculed. What a sad waste of human potential to spend a major chunk of your lifespan churning out digits that no one would have any use for even if they were correct. What I’ve been doing over the past several weeks may be just as foolish and futile: I have expended a nontrivial chunk of my own lifespan tracking down all things Shanks—reading his publications, casting about for scraps of biographical detail, and most of all trying to reconstruct his work and figure out where his calculations went awry.

The main results of this dubious effort are in my new American Scientist column (HTML, PDF). I have also made available some supplementary material, including an annotated reading list and an IPython notebook with code you are welcome to explore in case you too would like to devote a portion of your life to following the Pied Piper.

In this posting I merely want to tie up a couple of loose ends: one item that wouldn’t fit in the magazine (it would have required a multipage foldout), and another development I stumbled upon after the article was sent to the printer.

In one phase of his computation, Shanks faced the task of adding up 506 numbers, each with 709 decimal places. Half of these terms were positive and half were negative, so the problem would naturally decompose into two parts. But half of the addition problem is still a tableau of 253 × 709 = 179,377 digits. What would such an array of numbers look like, if you wrote them all down on one sheet of paper? I don’t have a sheet big enough to answer that question directly. But here’s a hint—in very tiny type, with scroll bars. These are the 253 positive terms in the series for arctan 1/5. Feel free to add up the columns and write the sum at the bottom.

 0.2000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000000000
0.0000640000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000000000
0.0000000568 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 888888889
0.0000000000 6301538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 846153846
0.0000000000 0007710117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 352941176
0.0000000000 0000009986 4380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 809523810
0.0000000000 0000000013 4217728000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000000000
0.0000000000 0000000000 0185127900 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 206896552
0.0000000000 0000000000 0000260301 0482424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 4242424242 424242424
0.0000000000 0000000000 0000000371 4566310054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 0540540540 5405405405 4054054054 054054054
0.0000000000 0000000000 0000000000 5363471355 0048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 8048780487 804878049
0.0000000000 0000000000 0000000000 0007818749 3530737777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 777777778
0.0000000000 0000000000 0000000000 0000011488 7745596186 1224489795 9183673469 3877551020 4081632653 0612244897 9591836734 6938775510 2040816326 5306122448 9795918367 3469387755 1020408163 2653061224 4897959183 6734693877 5510204081 6326530612 2448979591 8367346938 7755102040 8163265306 1224489795 9183673469 3877551020 4081632653 0612244897 9591836734 6938775510 2040816326 5306122448 9795918367 3469387755 1020408163 2653061224 4897959183 6734693877 5510204081 6326530612 2448979591 8367346938 7755102040 8163265306 1224489795 9183673469 3877551020 4081632653 0612244897 9591836734 6938775510 2040816326 5306122448 9795918367 3469387755 1020408163 2653061224 4897959183 6734693877 5510204081 6326530612 2448979591 8367346938 7755102040 8163265306 1224489795 9183673469 387755102
0.0000000000 0000000000 0000000000 0000000016 9947155749 8300377358 4905660377 3584905660 3773584905 6603773584 9056603773 5849056603 7735849056 6037735849 0566037735 8490566037 7358490566 0377358490 5660377358 4905660377 3584905660 3773584905 6603773584 9056603773 5849056603 7735849056 6037735849 0566037735 8490566037 7358490566 0377358490 5660377358 4905660377 3584905660 3773584905 6603773584 9056603773 5849056603 7735849056 6037735849 0566037735 8490566037 7358490566 0377358490 5660377358 4905660377 3584905660 3773584905 6603773584 9056603773 5849056603 7735849056 6037735849 0566037735 8490566037 7358490566 0377358490 5660377358 4905660377 3584905660 3773584905 6603773584 9056603773 5849056603 7735849056 6037735849 0566037735 8490566037 7358490566 0377358490 566037736
0.0000000000 0000000000 0000000000 0000000000 0252833663 2909752140 3508771929 8245614035 0877192982 4561403508 7719298245 6140350877 1929824561 4035087719 2982456140 3508771929 8245614035 0877192982 4561403508 7719298245 6140350877 1929824561 4035087719 2982456140 3508771929 8245614035 0877192982 4561403508 7719298245 6140350877 1929824561 4035087719 2982456140 3508771929 8245614035 0877192982 4561403508 7719298245 6140350877 1929824561 4035087719 2982456140 3508771929 8245614035 0877192982 4561403508 7719298245 6140350877 1929824561 4035087719 2982456140 3508771929 8245614035 0877192982 4561403508 7719298245 6140350877 1929824561 4035087719 2982456140 3508771929 8245614035 0877192982 4561403508 7719298245 6140350877 1929824561 4035087719 2982456140 3508771929 824561404
0.0000000000 0000000000 0000000000 0000000000 0000378007 0506907695 0032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 754098361
0.0000000000 0000000000 0000000000 0000000000 0000000567 5921253449 0928049230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 7692307692 3076923076 9230769230 769230769
0.0000000000 0000000000 0000000000 0000000000 0000000000 8555011744 3290674161 1594202898 5507246376 8115942028 9855072463 7681159420 2898550724 6376811594 2028985507 2463768115 9420289855 0724637681 1594202898 5507246376 8115942028 9855072463 7681159420 2898550724 6376811594 2028985507 2463768115 9420289855 0724637681 1594202898 5507246376 8115942028 9855072463 7681159420 2898550724 6376811594 2028985507 2463768115 9420289855 0724637681 1594202898 5507246376 8115942028 9855072463 7681159420 2898550724 6376811594 2028985507 2463768115 9420289855 0724637681 1594202898 5507246376 8115942028 9855072463 7681159420 2898550724 6376811594 2028985507 2463768115 9420289855 0724637681 1594202898 5507246376 8115942028 9855072463 7681159420 2898550724 6376811594 2028985507 246376812
0.0000000000 0000000000 0000000000 0000000000 0000000000 0012937990 3640264252 4300273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 3972602739 7260273972 6027397260 2739726027 397260274
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000019625 4191495881 3595302233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 6233766233 7662337662 3376623376 623376623
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000029 8500202373 9825122731 2987654320 9876543209 8765432098 7654320987 6543209876 5432098765 4320987654 3209876543 2098765432 0987654320 9876543209 8765432098 7654320987 6543209876 5432098765 4320987654 3209876543 2098765432 0987654320 9876543209 8765432098 7654320987 6543209876 5432098765 4320987654 3209876543 2098765432 0987654320 9876543209 8765432098 7654320987 6543209876 5432098765 4320987654 3209876543 2098765432 0987654320 9876543209 8765432098 7654320987 6543209876 5432098765 4320987654 3209876543 2098765432 0987654320 9876543209 8765432098 7654320987 6543209876 5432098765 4320987654 3209876543 2098765432 0987654320 9876543209 8765432098 7654320987 6543209876 5432098765 4320987654 3209876543 209876543
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0455125014 4431545128 3056037647 0588235294 1176470588 2352941176 4705882352 9411764705 8823529411 7647058823 5294117647 0588235294 1176470588 2352941176 4705882352 9411764705 8823529411 7647058823 5294117647 0588235294 1176470588 2352941176 4705882352 9411764705 8823529411 7647058823 5294117647 0588235294 1176470588 2352941176 4705882352 9411764705 8823529411 7647058823 5294117647 0588235294 1176470588 2352941176 4705882352 9411764705 8823529411 7647058823 5294117647 0588235294 1176470588 2352941176 4705882352 9411764705 8823529411 7647058823 5294117647 0588235294 1176470588 2352941176 4705882352 9411764705 8823529411 7647058823 5294117647 0588235294 1176470588 2352941176 4705882352 9411764705 882352941
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000695471 9321827979 0724669900 2247191011 2359550561 7977528089 8876404494 3820224719 1011235955 0561797752 8089887640 4494382022 4719101123 5955056179 7752808988 7640449438 2022471910 1123595505 6179775280 8988764044 9438202247 1910112359 5505617977 5280898876 4044943820 2247191011 2359550561 7977528089 8876404494 3820224719 1011235955 0561797752 8089887640 4494382022 4719101123 5955056179 7752808988 7640449438 2022471910 1123595505 6179775280 8988764044 9438202247 1910112359 5505617977 5280898876 4044943820 2247191011 2359550561 7977528089 8876404494 3820224719 1011235955 0561797752 8089887640 4494382022 4719101123 5955056179 7752808988 7640449438 2022471910 1123595505 6179775280 8988764044 943820225
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001064 8946574497 8948378419 2880860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 0537634408 6021505376 3440860215 053763441
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000001 6335703611 1885232151 6370110020 6185567010 3092783505 1546391752 5773195876 2886597938 1443298969 0721649484 5360824742 2680412371 1340206185 5670103092 7835051546 3917525773 1958762886 5979381443 2989690721 6494845360 8247422680 4123711340 2061855670 1030927835 0515463917 5257731958 7628865979 3814432989 6907216494 8453608247 4226804123 7113402061 8556701030 9278350515 4639175257 7319587628 8659793814 4329896907 2164948453 6082474226 8041237113 4020618556 7010309278 3505154639 1752577319 5876288659 7938144329 8969072164 9484536082 4742268041 2371134020 6185567010 3092783505 1546391752 5773195876 2886597938 1443298969 0721649484 5360824742 2680412371 1340206185 5670103092 7835051546 391752577
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0025101992 0837273148 8112283575 0019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 8019801980 1980198019 801980198
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000038633 1611498127 0556942333 5782887619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 7619047619 0476190476 1904761904 761904762
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000059 5446887446 6545445562 4957903900 1834862385 3211009174 3119266055 0458715596 3302752293 5779816513 7614678899 0825688073 3944954128 4403669724 7706422018 3486238532 1100917431 1926605504 5871559633 0275229357 7981651376 1467889908 2568807339 4495412844 0366972477 0642201834 8623853211 0091743119 2660550458 7155963302 7522935779 8165137614 6788990825 6880733944 9541284403 6697247706 4220183486 2385321100 9174311926 6055045871 5596330275 2293577981 6513761467 8899082568 8073394495 4128440366 9724770642 2018348623 8532110091 7431192660 5504587155 9633027522 9357798165 1376146788 9908256880 7339449541 2844036697 2477064220 1834862385 3211009174 3119266055 0458715596 3302752293 577981651
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0918990594 4309429431 0274271313 6300176991 1504424778 7610619469 0265486725 6637168141 5929203539 8230088495 5752212389 3805309734 5132743362 8318584070 7964601769 9115044247 7876106194 6902654867 2566371681 4159292035 3982300884 9557522123 8938053097 3451327433 6283185840 7079646017 6991150442 4778761061 9469026548 6725663716 8141592920 3539823008 8495575221 2389380530 9734513274 3362831858 4070796460 1769911504 4247787610 6194690265 4867256637 1681415929 2035398230 0884955752 2123893805 3097345132 7433628318 5840707964 6017699115 0442477876 1061946902 6548672566 3716814159 2920353982 3008849557 5221238938 0530973451 3274336283 1858407079 6460176991 1504424778 7610619469 026548673
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0001420115 3801120896 0780321267 1239684376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 7606837606 8376068376 0683760683 760683761
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000002197 0710674130 8408744430 9190546024 0925619834 7107438016 5289256198 3471074380 1652892561 9834710743 8016528925 6198347107 4380165289 2561983471 0743801652 8925619834 7107438016 5289256198 3471074380 1652892561 9834710743 8016528925 6198347107 4380165289 2561983471 0743801652 8925619834 7107438016 5289256198 3471074380 1652892561 9834710743 8016528925 6198347107 4380165289 2561983471 0743801652 8925619834 7107438016 5289256198 3471074380 1652892561 9834710743 8016528925 6198347107 4380165289 2561983471 0743801652 8925619834 7107438016 5289256198 3471074380 1652892561 9834710743 8016528925 6198347107 4380165289 2561983471 0743801652 8925619834 7107438016 5289256198 347107438
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000003 4028236692 0938463463 3746074317 6821145600 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000000000
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0052756956 1117734051 8812009417 5467939760 6201550387 5968992248 0620155038 7596899224 8062015503 8759689922 4806201550 3875968992 2480620155 0387596899 2248062015 5038759689 9224806201 5503875968 9922480620 1550387596 8992248062 0155038759 6899224806 2015503875 9689922480 6201550387 5968992248 0620155038 7596899224 8062015503 8759689922 4806201550 3875968992 2480620155 0387596899 2248062015 5038759689 9224806201 5503875968 9922480620 1550387596 8992248062 0155038759 6899224806 2015503875 9689922480 6201550387 5968992248 0620155038 7596899224 8062015503 8759689922 4806201550 3875968992 2480620155 0387596899 2248062015 5038759689 9224806201 5503875968 9922480620 155038760
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000081872 4491839851 9421675178 5246741223 8089624060 1503759398 4962406015 0375939849 6240601503 7593984962 4060150375 9398496240 6015037593 9849624060 1503759398 4962406015 0375939849 6240601503 7593984962 4060150375 9398496240 6015037593 9849624060 1503759398 4962406015 0375939849 6240601503 7593984962 4060150375 9398496240 6015037593 9849624060 1503759398 4962406015 0375939849 6240601503 7593984962 4060150375 9398496240 6015037593 9849624060 1503759398 4962406015 0375939849 6240601503 7593984962 4060150375 9398496240 6015037593 9849624060 1503759398 4962406015 0375939849 6240601503 7593984962 4060150375 9398496240 6015037593 9849624060 1503759398 4962406015 037593985
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000127 1712203383 3612649142 1751098587 8279310014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 5985401459 8540145985 4014598540 1459854014 598540146
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1977016418 3094524061 6451403126 3152615778 4056737588 6524822695 0354609929 0780141843 9716312056 7375886524 8226950354 6099290780 1418439716 3120567375 8865248226 9503546099 2907801418 4397163120 5673758865 2482269503 5460992907 8014184397 1631205673 7588652482 2695035460 9929078014 1843971631 2056737588 6524822695 0354609929 0780141843 9716312056 7375886524 8226950354 6099290780 1418439716 3120567375 8865248226 9503546099 2907801418 4397163120 5673758865 2482269503 5460992907 8014184397 1631205673 7588652482 2695035460 9929078014 1843971631 2056737588 6524822695 0354609929 0780141843 9716312056 7375886524 8226950354 6099290780 1418439716 3120567375 886524823
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0003075964 8549697411 2297044389 9675636069 7904022068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 827586207
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000004789 4217876038 9221831639 5870635889 7194051900 1342281879 1946308724 8322147651 0067114093 9597315436 2416107382 5503355704 6979865771 8120805369 1275167785 2348993288 5906040268 4563758389 2617449664 4295302013 4228187919 4630872483 2214765100 6711409395 9731543624 1610738255 0335570469 7986577181 2080536912 7516778523 4899328859 0604026845 6375838926 1744966442 9530201342 2818791946 3087248322 1476510067 1140939597 3154362416 1073825503 3557046979 8657718120 8053691275 1677852348 9932885906 0402684563 7583892617 4496644295 3020134228 1879194630 8724832214 7651006711 4093959731 5436241610 7382550335 5704697986 5771812080 5369127516 7785234899 328859060
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000007 4627330337 5665297285 3782846794 7458111516 1633464052 2875816993 4640522875 8169934640 5228758169 9346405228 7581699346 4052287581 6993464052 2875816993 4640522875 8169934640 5228758169 9346405228 7581699346 4052287581 6993464052 2875816993 4640522875 8169934640 5228758169 9346405228 7581699346 4052287581 6993464052 2875816993 4640522875 8169934640 5228758169 9346405228 7581699346 4052287581 6993464052 2875816993 4640522875 8169934640 5228758169 9346405228 7581699346 4052287581 6993464052 2875816993 4640522875 8169934640 5228758169 9346405228 7581699346 4052287581 6993464052 2875816993 4640522875 8169934640 5228758169 9346405228 7581699346 405228758
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0116361595 3288935444 4296853763 6646737233 7525910012 7388535031 8471337579 6178343949 0445859872 6114649681 5286624203 8216560509 5541401273 8853503184 7133757961 7834394904 4585987261 1464968152 8662420382 1656050955 4140127388 5350318471 3375796178 3439490445 8598726114 6496815286 6242038216 5605095541 4012738853 5031847133 7579617834 3949044585 9872611464 9681528662 4203821656 0509554140 1273885350 3184713375 7961783439 4904458598 7261146496 8152866242 0382165605 0955414012 7388535031 8471337579 6178343949 0445859872 6114649681 5286624203 8216560509 5541401273 8853503184 7133757961 7834394904 4585987261 1464968152 8662420382 1656050955 414012739
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000181552 9984261991 2027374966 8660407828 5286872724 1937888198 7577639751 5527950310 5590062111 8012422360 2484472049 6894409937 8881987577 6397515527 9503105590 0621118012 4223602484 4720496894 4099378881 9875776397 5155279503 1055900621 1180124223 6024844720 4968944099 3788819875 7763975155 2795031055 9006211180 1242236024 8447204968 9440993788 8198757763 9751552795 0310559006 2111801242 2360248447 2049689440 9937888198 7577639751 5527950310 5590062111 8012422360 2484472049 6894409937 8881987577 6397515527 9503105590 0621118012 4223602484 4720496894 4099378881 9875776397 5155279503 1055900621 1180124223 6024844720 4968944099 3788819875 776397516
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000283 4427417853 8723262132 0694829824 5856302414 6288019393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 9393939393 939393939
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 4427744605 4048656456 8216943402 8029029963 5352781284 7337278106 5088757396 4497041420 1183431952 6627218934 9112426035 5029585798 8165680473 3727810650 8875739644 9704142011 8343195266 2721893491 1242603550 2958579881 6568047337 2781065088 7573964497 0414201183 4319526627 2189349112 4260355029 5857988165 6804733727 8106508875 7396449704 1420118343 1952662721 8934911242 6035502958 5798816568 0473372781 0650887573 9644970414 2011834319 5266272189 3491124260 3550295857 9881656804 7337278106 5088757396 4497041420 1183431952 6627218934 9112426035 5029585798 8165680473 3727810650 8875739644 9704142011 8343195266 2721893491 1242603550 295857988
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0006920590 4121472582 1151710182 0584849998 2782655445 4300115606 9364161849 7109826589 5953757225 4335260115 6069364161 8497109826 5895953757 2254335260 1156069364 1618497109 8265895953 7572254335 2601156069 3641618497 1098265895 9537572254 3352601156 0693641618 4971098265 8959537572 2543352601 1560693641 6184971098 2658959537 5722543352 6011560693 6416184971 0982658959 5375722543 3526011560 6936416184 9710982658 9595375722 5433526011 5606936416 1849710982 6589595375 7225433526 0115606936 4161849710 9826589595 3757225433 5260115606 9364161849 7109826589 5953757225 4335260115 6069364161 8497109826 5895953757 2254335260 1156069364 161849711
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000010822 7086219342 4334998832 6462925912 3136916604 6739832045 1977401129 9435028248 5875706214 6892655367 2316384180 7909604519 7740112994 3502824858 7570621468 9265536723 1638418079 0960451977 4011299435 0282485875 7062146892 6553672316 3841807909 6045197740 1129943502 8248587570 6214689265 5367231638 4180790960 4519774011 2994350282 4858757062 1468926553 6723163841 8079096045 1977401129 9435028248 5875706214 6892655367 2316384180 7909604519 7740112994 3502824858 7570621468 9265536723 1638418079 0960451977 4011299435 0282485875 7062146892 6553672316 3841807909 6045197740 1129943502 8248587570 6214689265 5367231638 4180790960 451977401
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000016 9336523852 5844042384 9138421550 3777195440 7869816135 0011049723 7569060773 4806629834 2541436464 0883977900 5524861878 4530386740 3314917127 0718232044 1988950276 2430939226 5193370165 7458563535 9116022099 4475138121 5469613259 6685082872 9281767955 8011049723 7569060773 4806629834 2541436464 0883977900 5524861878 4530386740 3314917127 0718232044 1988950276 2430939226 5193370165 7458563535 9116022099 4475138121 5469613259 6685082872 9281767955 8011049723 7569060773 4806629834 2541436464 0883977900 5524861878 4530386740 3314917127 0718232044 1988950276 2430939226 5193370165 7458563535 9116022099 4475138121 5469613259 668508287
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0265080309 7713969917 1603840469 6269696625 9440643777 0416017297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 2972972972 9729729729 7297297297 297297297
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000415152 2311763677 7538596490 8942305979 9009494341 5767424027 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 0899470899 4708994708 9947089947 089947090
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000650 4768637706 4049366173 9836601830 7166945446 0564394140 0300103626 9430051813 4715025906 7357512953 3678756476 6839378238 3419689119 1709844559 5854922279 7927461139 8963730569 9481865284 9740932642 4870466321 2435233160 6217616580 3108808290 1554404145 0777202072 5388601036 2694300518 1347150259 0673575129 5336787564 7668393782 3834196891 1917098445 5958549222 7979274611 3989637305 6994818652 8497409326 4248704663 2124352331 6062176165 8031088082 9015544041 4507772020 7253886010 3626943005 1813471502 5906735751 2953367875 6476683937 8238341968 9119170984 4559585492 2279792746 1139896373 0569948186 5284974093 264248705
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000001 0196307387 4301413422 7133307323 0727173364 2321940532 5403110010 1522842639 5939086294 4162436548 2233502538 0710659898 4771573604 0609137055 8375634517 7664974619 2893401015 2284263959 3908629441 6243654822 3350253807 1065989847 7157360406 0913705583 7563451776 6497461928 9340101522 8426395939 0862944162 4365482233 5025380710 6598984771 5736040609 1370558375 6345177664 9746192893 4010152284 2639593908 6294416243 6548223350 2538071065 9898477157 3604060913 7055837563 4517766497 4619289340 1015228426 3959390862 9441624365 4822335025 3807106598 9847715736 0406091370 5583756345 1776649746 1928934010 1522842639 593908629
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0015989433 2762088584 6310410042 1305100522 6089850127 6410781394 4039800995 0248756218 9054726368 1592039800 9950248756 2189054726 3681592039 8009950248 7562189054 7263681592 0398009950 2487562189 0547263681 5920398009 9502487562 1890547263 6815920398 0099502487 5621890547 2636815920 3980099502 4875621890 5472636815 9203980099 5024875621 8905472636 8159203980 0995024875 6218905472 6368159203 9800995024 8756218905 4726368159 2039800995 0248756218 9054726368 1592039800 9950248756 2189054726 3681592039 8009950248 7562189054 7263681592 0398009950 2487562189 0547263681 5920398009 9502487562 1890547263 6815920398 009950249
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000025083 9109347744 8234992331 0709715708 9174197540 4929237596 5680015609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 7560975609 756097561
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000039 3661377349 5708234820 5004367400 7297842759 7764673636 0367191603 4449760765 5502392344 4976076555 0239234449 7607655502 3923444976 0765550239 2344497607 6555023923 4449760765 5502392344 4976076555 0239234449 7607655502 3923444976 0765550239 2344497607 6555023923 4449760765 5502392344 4976076555 0239234449 7607655502 3923444976 0765550239 2344497607 6555023923 4449760765 5502392344 4976076555 0239234449 7607655502 3923444976 0765550239 2344497607 6555023923 4449760765 5502392344 4976076555 0239234449 7607655502 3923444976 0765550239 2344497607 6555023923 4449760765 5502392344 4976076555 0239234449 760765550
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0618029880 6840210486 9670204039 7126950228 2576021158 9854022482 9680375586 8544600938 9671361502 3474178403 7558685446 0093896713 6150234741 7840375586 8544600938 9671361502 3474178403 7558685446 0093896713 6150234741 7840375586 8544600938 9671361502 3474178403 7558685446 0093896713 6150234741 7840375586 8544600938 9671361502 3474178403 7558685446 0093896713 6150234741 7840375586 8544600938 9671361502 3474178403 7558685446 0093896713 6150234741 7840375586 8544600938 9671361502 3474178403 7558685446 0093896713 6150234741 7840375586 8544600938 9671361502 3474178403 7558685446 0093896713 6150234741 784037559
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000970620 1997102044 8569509702 9342584629 6672359023 0920839865 7244032589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 8617511520 7373271889 4009216589 861751152
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001524 8838612641 7645897437 8681392513 0471514221 9584505500 4587308272 0144796380 0904977375 5656108597 2850678733 0316742081 4479638009 0497737556 5610859728 5067873303 1674208144 7963800904 9773755656 1085972850 6787330316 7420814479 6380090497 7375565610 8597285067 8733031674 2081447963 8009049773 7556561085 9728506787 3303167420 8144796380 0904977375 5656108597 2850678733 0316742081 4479638009 0497737556 5610859728 5067873303 1674208144 7963800904 9773755656 1085972850 6787330316 7420814479 6380090497 7375565610 8597285067 8733031674 2081447963 8009049773 7556561085 9728506787 3303167420 814479638
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000002 3964397037 4672353730 3734674401 7449487677 4617089249 5331653654 2443747555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 5555555555 555555556
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0037673287 9191624660 8861766300 3695553779 7549616384 8464276835 4389867900 0873362445 4148471615 7205240174 6724890829 6943231441 0480349344 9781659388 6462882096 0698689956 3318777292 5764192139 7379912663 7554585152 8384279475 9825327510 9170305676 8558951965 0655021834 0611353711 7903930131 0043668122 2707423580 7860262008 7336244541 4847161572 0524017467 2489082969 4323144104 8034934497 8165938864 6288209606 9868995633 1877729257 6419213973 7991266375 5458515283 8427947598 2532751091 7030567685 5895196506 5502183406 1135371179 0393013100 4366812227 0742358078 6026200873 3624454148 471615721
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000059242 4579123653 5439858159 5384352149 8064472873 1476726656 7918662865 2300085836 9098712446 3519313304 7210300429 1845493562 2317596566 5236051502 1459227467 8111587982 8326180257 5107296137 3390557939 9141630901 2875536480 6866952789 6995708154 5064377682 4034334763 9484978540 7725321888 4120171673 8197424892 7038626609 4420600858 3690987124 4635193133 0472103004 2918454935 6223175965 6652360515 0214592274 6781115879 8283261802 5751072961 3733905579 3991416309 0128755364 8068669527 8969957081 5450643776 8240343347 6394849785 4077253218 8841201716 7381974248 9270386266 0944206008 583690987
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000093 1881363279 7385704553 2568604584 3099065128 6375926260 3276456023 2749272033 7552742616 0337552742 6160337552 7426160337 5527426160 3375527426 1603375527 4261603375 5274261603 3755274261 6033755274 2616033755 2742616033 7552742616 0337552742 6160337552 7426160337 5527426160 3375527426 1603375527 4261603375 5274261603 3755274261 6033755274 2616033755 2742616033 7552742616 0337552742 6160337552 7426160337 5527426160 3375527426 1603375527 4261603375 5274261603 3755274261 6033755274 2616033755 2742616033 7552742616 0337552742 6160337552 7426160337 5527426160 3375527426 1603375527 426160338
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1466263124 2974143813 9692738650 8646071226 1277152494 7631379386 4067462755 0008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 1867219917 0124481327 8008298755 186721992
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0002307718 6152207468 7945573551 1174424183 9379381510 2121659836 2914222096 4829727346 9387755102 0408163265 3061224489 7959183673 4693877551 0204081632 6530612244 8979591836 7346938775 5102040816 3265306122 4489795918 3673469387 7551020408 1632653061 2244897959 1836734693 8775510204 0816326530 6122448979 5918367346 9387755102 0408163265 3061224489 7959183673 4693877551 0204081632 6530612244 8979591836 7346938775 5102040816 3265306122 4489795918 3673469387 7551020408 1632653061 2244897959 1836734693 8775510204 0816326530 6122448979 5918367346 9387755102 0408163265 3061224489 795918367
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000003633 0349283796 4970580983 2611969374 6188902480 1285267833 9591415160 8908486960 3212851405 6224899598 3935742971 8875502008 0321285140 5622489959 8393574297 1887550200 8032128514 0562248995 9839357429 7188755020 0803212851 4056224899 5983935742 9718875502 0080321285 1405622489 9598393574 2971887550 2008032128 5140562248 9959839357 4297188755 0200803212 8514056224 8995983935 7429718875 5020080321 2851405622 4899598393 5742971887 5502008032 1285140562 2489959839 3574297188 7550200803 2128514056 2248995983 9357429718 8755020080 3212851405 6224899598 3935742971 8875502008 032128514
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000005 7209530255 5910056440 5681077504 3432733829 0438071346 6602739999 2098771309 1106403162 0553359683 7944664031 6205533596 8379446640 3162055335 9683794466 4031620553 3596837944 6640316205 5335968379 4466403162 0553359683 7944664031 6205533596 8379446640 3162055335 9683794466 4031620553 3596837944 6640316205 5335968379 4466403162 0553359683 7944664031 6205533596 8379446640 3162055335 9683794466 4031620553 3596837944 6640316205 5335968379 4466403162 0553359683 7944664031 6205533596 8379446640 3162055335 9683794466 4031620553 3596837944 6640316205 5335968379 4466403162 055335968
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0090110575 2819581287 3429648638 8083974983 0894822300 7047785171 7964208492 7078910007 7821011673 1517509727 6264591439 6887159533 0739299610 8949416342 4124513618 6770428015 5642023346 3035019455 2529182879 3774319066 1478599221 7898832684 8249027237 3540856031 1284046692 6070038910 5058365758 7548638132 2957198443 5797665369 6498054474 7081712062 2568093385 2140077821 0116731517 5097276264 5914396887 1595330739 2996108949 4163424124 5136186770 4280155642 0233463035 0194552529 1828793774 3190661478 5992217898 8326848249 0272373540 8560311284 0466926070 0389105058 365758755
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000141967 3124748704 3117066174 4071954523 0384919352 2224130456 9449911427 3264424704 1961685823 7547892720 3065134099 6168582375 4789272030 6513409961 6858237547 8927203065 1340996168 5823754789 2720306513 4099616858 2375478927 2030651340 9961685823 7547892720 3065134099 6168582375 4789272030 6513409961 6858237547 8927203065 1340996168 5823754789 2720306513 4099616858 2375478927 2030651340 9961685823 7547892720 3065134099 6168582375 4789272030 6513409961 6858237547 8927203065 1340996168 5823754789 2720306513 4099616858 2375478927 2030651340 9961685823 7547892720 306513410
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000223 7190554320 9770587497 4884243200 7880032989 8924078478 7880008615 1398170655 6848012075 4716981132 0754716981 1320754716 9811320754 7169811320 7547169811 3207547169 8113207547 1698113207 5471698113 2075471698 1132075471 6981132075 4716981132 0754716981 1320754716 9811320754 7169811320 7547169811 3207547169 8113207547 1698113207 5471698113 2075471698 1132075471 6981132075 4716981132 0754716981 1320754716 9811320754 7169811320 7547169811 3207547169 8113207547 1698113207 5471698113 2075471698 1132075471 6981132075 4716981132 0754716981 1320754716 9811320754 716981132
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 3526278048 4464471111 8919520145 4168554401 4413174012 1747439121 3859452878 6543135900 0743494423 7918215613 3828996282 5278810408 9219330855 0185873605 9479553903 3457249070 6319702602 2304832713 7546468401 4869888475 8364312267 6579925650 5576208178 4386617100 3717472118 9591078066 9144981412 6394052044 6096654275 0929368029 7397769516 7286245353 1598513011 1524163568 7732342007 4349442379 1821561338 2899628252 7881040892 1933085501 8587360594 7955390334 5724907063 1970260223 0483271375 4646840148 6988847583 6431226765 7992565055 7620817843 8661710037 174721190
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0005559377 5533016514 1632172020 0387817383 9354591319 0287179845 4133498566 0035634306 5618754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 7545787545 7875457875 4578754578 754578755
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000008766 5563728597 8822617085 6980250536 5823213305 0113781372 4060525314 7062467743 9173510007 2202166064 9819494584 8375451263 5379061371 8411552346 5703971119 1335740072 2021660649 8194945848 3754512635 3790613718 4115523465 7039711191 3357400722 0216606498 1949458483 7545126353 7906137184 1155234657 0397111913 3574007220 2166064981 9494584837 5451263537 9061371841 1552346570 3971119133 5740072202 1660649819 4945848375 4512635379 0613718411 5523465703 9711191335 7400722021 6606498194 9458483754 5126353790 6137184115 5234657039 7111913357 4007220216 606498195
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000013 8268248556 9913929480 0440617960 9886593765 6113204369 7660155248 4678925571 9007265835 0007117437 7224199288 2562277580 0711743772 2419928825 6227758007 1174377224 1992882562 2775800711 7437722419 9288256227 7580071174 3772241992 8825622775 8007117437 7224199288 2562277580 0711743772 2419928825 6227758007 1174377224 1992882562 2775800711 7437722419 9288256227 7580071174 3772241992 8825622775 8007117437 7224199288 2562277580 0711743772 2419928825 6227758007 1174377224 1992882562 2775800711 7437722419 9288256227 7580071174 3772241992 8825622775 800711744
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0218124226 4955204570 8569396848 5447736886 1649784198 8934975458 9463549631 3583388304 2786537543 8596491228 0701754385 9649122807 0175438596 4912280701 7543859649 1228070175 4385964912 2807017543 8596491228 0701754385 9649122807 0175438596 4912280701 7543859649 1228070175 4385964912 2807017543 8596491228 0701754385 9649122807 0175438596 4912280701 7543859649 1228070175 4385964912 2807017543 8596491228 0701754385 9649122807 0175438596 4912280701 7543859649 1228070175 4385964912 2807017543 8596491228 0701754385 9649122807 0175438596 4912280701 754385965
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000344168 3296953540 7761618148 6606796429 1352637721 7809530618 7172302348 2072643685 3520798135 3633217993 0795847750 8650519031 1418685121 1072664359 8615916955 0173010380 6228373702 4221453287 1972318339 1003460207 6124567474 0484429065 7439446366 7820069204 1522491349 4809688581 3148788927 3356401384 0830449826 9896193771 6262975778 5467128027 6816608996 5397923875 4325259515 5709342560 5536332179 9307958477 5086505190 3114186851 2110726643 5986159169 5501730103 8062283737 0242214532 8719723183 3910034602 0761245674 7404844290 6574394463 667820069
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000543 1516575123 9496754171 4400706425 4908946961 3738959477 6726786595 9242697578 2938798693 0300068259 3856655290 1023890784 9829351535 8361774744 0273037542 6621160409 5563139931 7406143344 7098976109 2150170648 4641638225 2559726962 4573378839 5904436860 0682593856 6552901023 8907849829 3515358361 7747440273 0375426621 1604095563 1399317406 1433447098 9761092150 1706484641 6382252559 7269624573 3788395904 4368600682 5938566552 9010238907 8498293515 3583617747 4402730375 4266211604 0955631399 3174061433 4470989761 0921501706 4846416382 252559727
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 8573383738 7815070970 8941788051 0850846176 2137780552 8725015210 6268326520 6218995652 6171463542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 4208754208 7542087542 0875420875 420875421
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0013535122 7663421002 4098635008 9098858212 9665673997 7366279492 4521258168 6465100553 9339656742 4425249169 4352159468 4385382059 8006644518 2724252491 6943521594 6843853820 5980066445 1827242524 9169435215 9468438538 2059800664 4518272425 2491694352 1594684385 3820598006 6445182724 2524916943 5215946843 8538205980 0664451827 2425249169 4352159468 4385382059 8006644518 2724252491 6943521594 6843853820 5980066445 1827242524 9169435215 9468438538 2059800664 4518272425 2491694352 1594684385 3820598006 6445182724 2524916943 5215946843 853820598
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000021372 1807353126 4090511156 1321016426 6077531110 1551146230 1756227665 3574431450 5795888454 7120010491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 1967213114 7540983606 5573770491 8032786885 2459016393 4426229508 196721311
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000033 7528291224 3549761066 1624351637 5928303673 7273647117 6709569705 8274900720 2196855642 6080888560 2588996763 7540453074 4336569579 2880258899 6763754045 3074433656 9579288025 8899676375 4045307443 3656957928 8025889967 6375404530 7443365695 7928802588 9967637540 4530744336 5695792880 2588996763 7540453074 4336569579 2880258899 6763754045 3074433656 9579288025 8899676375 4045307443 3656957928 8025889967 6375404530 7443365695 7928802588 9967637540 4530744336 5695792880 2588996763 7540453074 4336569579 2880258899 6763754045 307443366
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0533143729 0138054319 0770310157 9860252247 0808013070 7187748278 7941249555 6424172924 6955445339 6300063897 7635782747 6038338658 1469648562 3003194888 1789137380 1916932907 3482428115 0159744408 9456869009 5846645367 4121405750 7987220447 2843450479 2332268370 6070287539 9361022364 2172523961 6613418530 3514376996 8051118210 8626198083 0670926517 5718849840 2555910543 1309904153 3546325878 5942492012 7795527156 5495207667 7316293929 7124600638 9776357827 4760383386 5814696485 6230031948 8817891373 8019169329 0734824281 150159744
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000842266 1813568257 4226933032 5776637269 1651043069 2297033515 2618930529 2664533835 3333991441 7226710006 3091482649 8422712933 7539432176 6561514195 5835962145 1104100946 3722397476 3406940063 0914826498 4227129337 5394321766 5615141955 8359621451 1041009463 7223974763 4069400630 9148264984 2271293375 3943217665 6151419558 3596214511 0410094637 2239747634 0694006309 1482649842 2712933753 9432176656 1514195583 5962145110 4100946372 2397476340 6940063091 4826498422 7129337539 4321766561 5141955835 9621451104 1009463722 397476341
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001330 8330441874 8291738007 5829389129 0433661710 4196676185 0434293836 6493580845 9852993770 9010646066 4024922118 3800623052 9595015576 3239875389 4080996884 7352024922 1183800623 0529595015 5763239875 3894080996 8847352024 9221183800 6230529595 0155763239 8753894080 9968847352 0249221183 8006230529 5950155763 2398753894 0809968847 3520249221 1838006230 5295950155 7632398753 8940809968 8473520249 2211838006 2305295950 1557632398 7538940809 9688473520 2492211838 0062305295 9501557632 3987538940 8099688473 5202492211 838006231
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000002 1031256969 0674223497 3559833766 9436205318 9368355116 5804255548 6600095701 9153786146 4637993439 4452440615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 1538461538 4615384615 3846153846 153846154
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0033240892 4739059562 9745444114 1605187923 3004472780 1234705814 2410431762 2036352488 7421312330 0562660392 4620060790 2735562310 0303951367 7811550151 9756838905 7750759878 4194528875 3799392097 2644376899 6960486322 1884498480 2431610942 2492401215 8054711246 2006079027 3556231003 0395136778 1155015197 5683890577 5075987841 9452887537 9939209726 4437689969 6048632218 8449848024 3161094224 9240121580 5471124620 0607902735 5623100303 9513677811 5501519756 8389057750 7598784194 5288753799 3920972644 3768996960 486322188
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000052546 5639587510 3722216221 5666489402 1706467130 5449159006 4282777331 1700420228 0782998735 1667916469 7615375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 3753753753 7537537537 5375375375 375375375
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000083 0765853923 5178822542 4416935031 3159540669 6948199738 6309346776 7419804106 5208312111 9468749750 4305310005 9347181008 9020771513 3531157270 0296735905 0445103857 5667655786 3501483679 5252225519 2878338278 9317507418 3976261127 5964391691 3946587537 0919881305 6379821958 4569732937 6854599406 5281899109 7922848664 6884272997 0326409495 5489614243 3234421364 9851632047 4777448071 2166172106 8249258160 2373887240 3560830860 5341246290 8011869436 2017804154 3026706231 4540059347 1810089020 7715133531 157270030
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1313633279 8696798888 8999547247 4160866933 5164206654 8359818181 1789421578 8100763407 3042866715 1478948455 0005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 8005865102 6392961876 8328445747 800586510
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0002077444 3985417314 4167646820 2266383098 5476132148 2634159758 8395687850 4736196453 6667397646 6080715693 5566617971 0144927536 2318840579 7101449275 3623188405 7971014492 7536231884 0579710144 9275362318 8405797101 4492753623 1884057971 0144927536 2318840579 7101449275 3623188405 7971014492 7536231884 0579710144 9275362318 8405797101 4492753623 1884057971 0144927536 2318840579 7101449275 3623188405 7971014492 7536231884 0579710144 9275362318 8405797101 4492753623 1884057971 0144927536 2318840579 710144928
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000003285 8146360889 2766706421 3317051700 4882472277 7817805318 5870471116 6001760402 4138224651 8651222220 8104391900 0573065902 5787965616 0458452722 0630372492 8366762177 6504297994 2693409742 1203438395 4154727793 6962750716 3323782234 9570200573 0659025787 9656160458 4527220630 3724928366 7621776504 2979942693 4097421203 4383954154 7277936962 7507163323 7822349570 2005730659 0257879656 1604584527 2206303724 9283667621 7765042979 9426934097 4212034383 9541547277 9369627507 1633237822 3495702005 730659026
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000005 1977305744 8174846823 9820725897 0808856579 3765250601 9543881221 1657494002 8575294007 4946217684 1022678448 8300056657 2237960339 9433427762 0396600566 5722379603 3994334277 6203966005 6657223796 0339943342 7762039660 0566572237 9603399433 4277620396 6005665722 3796033994 3342776203 9660056657 2237960339 9433427762 0396600566 5722379603 3994334277 6203966005 6657223796 0339943342 7762039660 0566572237 9603399433 4277620396 6005665722 3796033994 3342776203 9660056657 2237960339 9433427762 039660057
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0082231883 1503443006 9641038504 1643453339 4846830850 2520555697 7975193704 7977981305 4774032559 5181539520 4142834375 3501400560 2240896358 5434173669 4677871148 4593837535 0140056022 4089635854 3417366946 7787114845 9383753501 4005602240 8963585434 1736694677 8711484593 8375350140 0560224089 6358543417 3669467787 1148459383 7535014005 6022408963 5854341736 6946778711 4845938375 3501400560 2240896358 5434173669 4677871148 4593837535 0140056022 4089635854 3417366946 7787114845 9383753501 400560224
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000130113 1624805447 7716839227 6830434184 3421430775 0184597621 6470413935 3024390645 2124340519 1080242923 4738521847 6321329639 8891966759 0027700831 0249307479 2243767313 0193905817 1745152354 5706371191 1357340720 2216066481 9944598337 9501385041 5512465373 9612188365 6509695290 8587257617 7285318559 5567867036 0110803324 0997229916 8975069252 0775623268 6980609418 2825484764 5429362880 8864265927 9778393351 8005540166 2049861495 8448753462 6038781163 4349030470 9141274238 2271468144 044321330
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000205 8996236952 4017340400 9257800708 9996603338 1250703078 3185296742 7096731200 1004786353 6543473830 9934205397 8608008767 1232876712 3287671232 8767123287 6712328767 1232876712 3287671232 8767123287 6712328767 1232876712 3287671232 8767123287 6712328767 1232876712 3287671232 8767123287 6712328767 1232876712 3287671232 8767123287 6712328767 1232876712 3287671232 8767123287 6712328767 1232876712 3287671232 8767123287 6712328767 1232876712 3287671232 8767123287 6712328767 1232876712 328767123
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 3258682391 2742618229 7944191207 6261674841 0554608158 8014580523 8421204582 7123541450 4892524088 6647700747 8383488013 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 8753387533 875338753
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0005157978 7769660165 6913421583 0803820088 0057885899 8846015571 9605265091 6504916190 0063508166 8148731374 0523425231 4300053619 3029490616 6219839142 0911528150 1340482573 7265415549 5978552278 8203753351 2064343163 5388739946 3806970509 3833780160 8579088471 8498659517 4262734584 4504021447 7211796246 6487935656 8364611260 0536193029 4906166219 8391420911 5281501340 4825737265 4155495978 5522788203 7533512064 3431635388 7399463806 9705093833 7801608579 0884718498 6595174262 734584450
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000008165 2035387090 6813914933 6890779097 6883868823 0903213002 9001221279 0628567994 6477341914 2545228548 7640170769 7032292498 6737400530 5039787798 4084880636 6047745358 0901856763 9257294429 7082228116 7108753315 6498673740 0530503978 7798408488 0636604774 5358090185 6763925729 4429708222 8116710875 3315649867 3740053050 3978779840 8488063660 4774535809 0185676392 5729442970 8222811671 0875331564 9867374005 3050397877 9840848806 3660477453 5809018567 6392572944 2970822281 167108753
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000012 9271673872 6852992528 8398982986 7499360497 8322521832 2397620831 1693719559 6177730006 9361635910 3953135304 4836876322 4020997375 3280839895 0131233595 8005249343 8320209973 7532808398 9501312335 9580052493 4383202099 7375328083 9895013123 3595800524 9343832020 9973753280 8398950131 2335958005 2493438320 2099737532 8083989501 3123359580 0524934383 2020997375 3280839895 0131233595 8005249343 8320209973 7532808398 9501312335 9580052493 4383202099 7375328083 9895013123 359580052
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0204685746 4747765153 8846254597 4772019766 6479962102 1011256077 0355346731 0221895958 9928007411 0414986574 7574370285 2107436883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 8311688311 6883116883 1168831168 831168831
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000324129 6139549160 2436836228 3600153121 2757202202 1976717053 3250742657 0342150868 6651402705 8231402652 2623408257 3153979900 0514138817 4807197943 4447300771 2082262210 7969151670 9511568123 3933161953 7275064267 3521850899 7429305912 5964010282 7763496143 9588688946 0154241645 2442159383 0334190231 3624678663 2390745501 2853470437 0179948586 1182519280 2056555269 9228791773 7789203084 8329048843 1876606683 8046272493 5732647814 9100257069 4087403598 9717223650 385604113
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000513 3289356883 9626861798 6476622736 1391399701 4011395187 5165127893 7143462989 0601963951 7661687595 4472470252 6701096786 4880203562 3409669211 1959287531 8066157760 8142493638 6768447837 1501272264 6310432569 9745547073 7913486005 0890585241 7302798982 1882951653 9440203562 3409669211 1959287531 8066157760 8142493638 6768447837 1501272264 6310432569 9745547073 7913486005 0890585241 7302798982 1882951653 9440203562 3409669211 1959287531 8066157760 8142493638 676844784
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 8130509691 7094099170 5264222922 8626808342 9023703690 9166508394 0039485666 3161739332 3246109997 8841743801 7503248487 8069110005 0377833753 1486146095 7178841309 8236775818 6397984886 6498740554 1561712846 3476070528 9672544080 6045340050 3778337531 4861460957 1788413098 2367758186 3979848866 4987405541 5617128463 4760705289 6725440806 0453400503 7783375314 8614609571 7884130982 3677581863 9798488664 9874055415 6171284634 7607052896 7254408060 4534005037 783375315
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0012879051 7610319630 4067790110 7246942016 6070114355 5722450289 6056641100 4868928520 7578368900 4255761285 9423237065 9338028675 0004987531 1720698254 3640897755 6109725685 7855361596 0099750623 4413965087 2817955112 2194513715 7107231920 1995012468 8279301745 6359102244 3890274314 2144638403 9900249376 5586034912 7182044887 7805486284 2892768079 8004987531 1720698254 3640897755 6109725685 7855361596 0099750623 4413965087 2817955112 2194513715 7107231920 199501247
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000020402 9619996990 3083184923 7902246019 7477902679 9272472899 0266938125 7532651616 1454771065 3987976534 4223718392 3493904393 0811859753 0864197530 8641975308 6419753086 4197530864 1975308641 9753086419 7530864197 5308641975 3086419753 0864197530 8641975308 6419753086 4197530864 1975308641 9753086419 7530864197 5308641975 3086419753 0864197530 8641975308 6419753086 4197530864 1975308641 9753086419 7530864197 5308641975 3086419753 0864197530 8641975308 641975309
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000032 3254752464 6692415412 7887923362 8870331738 2312783771 2434457153 8031494274 4431840321 8835658212 2110163739 6534044132 1435613900 0488997555 0122249388 7530562347 1882640586 7970660146 6992665036 6748166259 1687041564 7921760391 1980440097 7995110024 4498777506 1124694376 5281173594 1320293398 5330073349 6332518337 4083129584 3520782396 0880195599 0220048899 7555012224 9388753056 2347188264 0586797066 0146699266 5036674816 6259168704 1564792176 039119804
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0512198329 3290061783 6468014181 7159159189 0786889795 8835392515 1608929321 6784397569 7497351222 1163944046 3710149826 8282681515 0997384987 8934624697 3365617433 4140435835 3510895883 7772397094 4309927360 7748184019 3704600484 2615012106 5375302663 4382566585 9564164648 9104116222 7602905569 0072639225 1815980629 5399515738 4987893462 4697336561 7433414043 5835351089 5883777239 7094430992 7360774818 4019370460 0484261501 2106537530 2663438256 658595642
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000811656 2494498975 6034383845 9742874753 9379932798 0266662895 3813388922 0521945155 6692820742 6560539890 2298368506 0085566417 1347192019 1846522781 7745803357 3141486810 5515587529 9760191846 5227817745 8033573141 4868105515 5875299760 1918465227 8177458033 5731414868 1055155875 2997601918 4652278177 4580335731 4148681055 1558752997 6019184652 2781774580 3357314148 6810551558 7529976019 1846522781 7745803357 3141486810 5515587529 9760191846 522781775
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001286 3112817885 3124052591 2162328850 4413953185 6599605504 7121102833 9401264232 3227138836 1033256988 6336374041 4898073847 7755636215 0004750593 8242280285 0356294536 8171021377 6722090261 2826603325 4156769596 1995249406 1757719714 9643705463 1828978622 3277909738 7173396674 5843230403 8004750593 8242280285 0356294536 8171021377 6722090261 2826603325 4156769596 1995249406 1757719714 9643705463 1828978622 3277909738 7173396674 5843230403 800475059
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000002 0387277162 6530504380 7657488340 5027231150 2726130574 7481742999 6916283980 2044531295 8117628239 7824555721 3043471394 2208508697 7751160470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 705882353
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0032315497 6004756976 6408408140 0269040832 5925067060 1184819546 0134972198 3835408581 0749463839 6341540088 3228134173 7717253580 2202496011 9347319347 3193473193 4731934731 9347319347 3193473193 4731934731 9347319347 3193473193 4731934731 9347319347 3193473193 4731934731 9347319347 3193473193 4731934731 9347319347 3193473193 4731934731 9347319347 3193473193 4731934731 9347319347 3193473193 4731934731 9347319347 3193473193 4731934731 934731935
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000051227 1537020011 9832939333 3656223255 4907424863 8107744249 7376066154 5426569571 4781751575 0104500769 3224590742 0251198898 0541874811 2300046189 3764434180 1385681293 3025404157 0438799076 2124711316 3972286374 1339491916 8591224018 4757505773 6720554272 5173210161 6628175519 6304849884 5265588914 5496535796 7667436489 6073903002 3094688221 7090069284 0646651270 2078521939 9538106235 5658198614 3187066974 5958429561 2009237875 288683603
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000081 2132084319 1402792357 8264432566 2961189620 0129247243 0577075832 1095358186 2680093358 1033575281 7688480541 1122997781 6855193159 8837989638 4439359267 7345537757 4370709382 1510297482 8375286041 1899313501 1441647597 2540045766 5903890160 1830663615 5606407322 6544622425 6292906178 4897025171 6247139588 1006864988 5583524027 4599542334 0961098398 1693363844 3935926773 4553775743 7070938215 1029748283 7528604118 9931350114 416475973
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1287625291 0565598259 4480549866 7808486312 4315488366 6039590683 4961700395 5642598460 9432786113 0090992711 0982896145 2357245240 4929495515 5446712018 1405895691 6099773242 6303854875 2834467120 1814058956 9160997732 4263038548 7528344671 2018140589 5691609977 3242630385 4875283446 7120181405 8956916099 7732426303 8548752834 4671201814 0589569160 9977324263 0385487528 3446712018 1405895691 6099773242 6303854875 2834467120 181405896
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0002041681 8098191204 7907113429 1820237456 0495543839 5314061876 8230896574 8294609926 9079586008 7221106077 4313504565 2136591623 0150378094 4624007191 0112359550 5617977528 0898876404 4943820224 7191011235 9550561797 7528089887 6404494382 0224719101 1235955056 1797752808 9887640449 4382022471 9101123595 5056179775 2808988764 0449438202 2471910112 3595505617 9775280898 8764044943 8202247191 0112359550 5617977528 0898876404 494382022
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000003237 5889723635 0508039787 8877006701 7120429459 2733771964 4906415141 1164333546 2538050479 3724056631 4635437005 0120982746 6271418862 3680561900 0445434298 4409799554 5657015590 2004454342 9844097995 5456570155 9020044543 4298440979 9554565701 5590200445 4342984409 7995545657 0155902004 4543429844 0979955456 5701559020 0445434298 4409799554 5657015590 2004454342 9844097995 5456570155 9020044543 4298440979 9554565701 559020045
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000005 1344015844 2812814516 0786280886 5662470897 4029624595 5392716928 5593334342 6481870497 7138656428 8830980348 6631361841 0710307698 2576715487 1280176600 4415011037 5275938189 8454746136 8653421633 5540838852 0971302428 2560706401 7660044150 1103752759 3818984547 4613686534 2163355408 3885209713 0242825607 0640176600 4415011037 5275938189 8454746136 8653421633 5540838852 0971302428 2560706401 7660044150 1103752759 381898455
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0081431384 4287419536 0202489926 4476656802 8587257487 7611528755 4263015423 9202437767 4764617282 6688281607 3451007464 0315279717 7673898475 6784910004 3763676148 7964989059 0809628008 7527352297 5929978118 1619256017 5054704595 1859956236 3238512035 0109409190 3719912472 6477024070 0218818380 7439824945 2954048140 0437636761 4879649890 5908096280 0875273522 9759299781 1816192560 1750547045 9518599562 3632385120 350109409
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000129159 7143043299 7065900348 4473287053 0460960960 2495682320 6637043637 4793285949 0142706915 7420551998 9398931402 7281992478 3723026365 6516618495 0004338394 7939262472 8850325379 6095444685 4663774403 4707158351 4099783080 2603036876 3557483731 0195227765 7266811279 8264642082 4295010845 9869848156 1822125813 4490238611 7136659436 0086767895 8785249457 7006507592 1908893709 3275488069 4143167028 1995661605 206073753
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000204 8778607975 7803517866 8753005368 8824232268 3963012505 9778958028 7892575328 4214901635 7441964299 2568130433 9841985371 2947256138 1662465930 7568006881 7204301075 2688172043 0107526881 7204301075 2688172043 0107526881 7204301075 2688172043 0107526881 7204301075 2688172043 0107526881 7204301075 2688172043 0107526881 7204301075 2688172043 0107526881 7204301075 2688172043 0107526881 7204301075 2688172043 010752688
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 3250088026 2984660591 2429748904 0393358696 8180956250 1160928666 8942840247 4293082914 3219481495 6047911916 9357873000 2478534666 4537221481 1353583362 7292110874 2004264392 3240938166 3113006396 5884861407 2494669509 5948827292 1108742004 2643923240 9381663113 0063965884 8614072494 6695095948 8272921108 7420042643 9232409381 6631130063 9658848614 0724946695 0959488272 9211087420 0426439232 4093816631 130063966
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0005156165 0210028941 4538450916 6545266757 6450154311 9874991883 4495506780 8069153339 1520236572 7575091655 6077560566 3834798926 8788728818 1806426488 2777843551 7970401691 3319238900 6342494714 5877378435 5179704016 9133192389 0063424947 1458773784 3551797040 1691331923 8900634249 4714587737 8435517970 4016913319 2389006342 4947145877 3784355179 7040169133 1923890063 4249471458 7737843551 7970401691 331923890
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000008180 6827838469 3980970020 2384153999 7530099532 0404866653 7890872535 7059007830 7066462295 6776999642 2745575467 1627652360 2261256415 0306061013 7658881073 3752620545 0733752620 5450733752 6205450733 7526205450 7337526205 4507337526 2054507337 5262054507 3375262054 5073375262 0545073375 2620545073 3752620545 0733752620 5450733752 6205450733 7526205450 7337526205 4507337526 2054507337 5262054507 337526205
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000012 9802434524 5734815543 2836714316 6998160024 8707968802 9733302107 9402302359 2024850569 6965929742 4677420838 2464420424 7012985012 2576951321 7582247937 7031185031 1850311850 3118503118 5031185031 1850311850 3118503118 5031185031 1850311850 3118503118 5031185031 1850311850 3118503118 5031185031 1850311850 3118503118 5031185031 1850311850 3118503118 5031185031 1850311850 3118503118 5031185031 185031185
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0205971038 3713644358 8497136361 5219220173 1033826088 2283881957 3242696931 7435922319 5844092741 2985632459 9528133232 9048456068 5885227261 3138662470 1296006597 9381443298 9690721649 4845360824 7422680412 3711340206 1855670103 0927835051 5463917525 7731958762 8865979381 4432989690 7216494845 3608247422 6804123711 3402061855 6701030927 8350515463 9175257731 9587628865 9793814432 9896907216 494845361
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000326857 9259226560 3731439218 4387341748 1683603781 2770250086 7052835036 4397403426 8302607394 6160484357 4415611107 1344256854 6200913980 2756228225 1059725360 1635991820 0408997955 0102249488 7525562372 1881390593 0470347648 2617586912 0654396728 0163599182 0040899795 5010224948 8752556237 2188139059 3047034764 8261758691 2065439672 8016359918 2004089979 5501022494 8875255623 7218813905 930470348
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000518 7294954196 4722838697 8590394840 1293446757 8062295017 5818219347 1247676528 2781987748 5305225077 0005320028 6617078593 4341881531 6392852886 3353405941 4235131845 8417849898 5801217038 5395537525 3549695740 3651115618 6612576064 9087221095 3346855983 7728194726 1663286004 0567951318 4584178498 9858012170 3853955375 2535496957 4036511156 1866125760 6490872210 9533468559 8377281947 261663286
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 8232873762 3142401157 2733827169 9193038774 2506188205 8548662799 6398632127 2545389199 0664806361 2489698584 3835178172 2223357899 7025293123 4542078806 0494866543 2595573440 6438631790 7444668008 0482897384 3058350100 6036217303 8229376257 5452716297 7867203219 3158953722 3340040241 4486921529 1750503018 1086519114 6881287726 3581488933 6016096579 4768611670 0201207243 4607645875 251509054
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0013067427 5764317040 7190894090 5499397813 2400880081 5594726340 6352766850 8335185815 3514923461 1147009597 4337360745 8733556914 5745378269 6442530660 8115180674 4015968063 8722554890 2195608782 4351297405 1896207584 8303393213 5728542914 1716566866 2674650698 6027944111 7764471057 8842315369 2614770459 0818363273 4530938123 7524950099 8003992015 9680638722 5548902195 6087824351 297405190
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000020742 2771193418 8907731130 1049204588 6873133753 4126298480 4646598768 1339171257 3259123886 4285733550 2837680848 2651328354 8964810881 6256372935 7974386393 7200006336 6336633663 3663366336 6336633663 3663366336 6336633663 3663366336 6336633663 3663366336 6336633663 3663366336 6336633663 3663366336 6336633663 3663366336 6336633663 3663366336 6336633663 3663366336 6336633663 366336634
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000032 9268367631 2032686535 8607559444 6122973461 8325292827 4503250396 4727941159 8741047882 3187314091 7654799304 7650259868 9798749640 8494175146 0356077218 4796183900 0392927308 4479371316 3064833005 8939096267 1905697445 9724950884 0864440078 5854616895 8742632612 9666011787 8192534381 1394891944 9901768172 8880157170 9233791748 5265225933 2023575638 5068762278 9783889980 353634578
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0522721556 7229082689 8924074067 0793221344 2247883271 8882564224 4110866345 5762211909 1527492687 3222881225 2510733667 9089614233 3479683540 4267668595 1008378191 3609980506 8226120857 6998050682 2612085769 9805068226 1208576998 0506822612 0857699805 0682261208 5769980506 8226120857 6998050682 2612085769 9805068226 1208576998 0506822612 0857699805 0682261208 5769980506 822612086
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000829883 6629751704 2008968820 1049491443 0935003989 5349436767 3411926884 1323964455 5803338038 6803755011 4307274294 3803088501 6687272967 6015131339 2318351405 8403657779 4970986460 3481624758 2205029013 5396518375 2417794970 9864603481 6247582205 0290135396 5183752417 7949709864 6034816247 5822050290 1353965183 7524177949 7098646034 8162475822 0502901353 9651837524 177949710
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001317 6195125010 7672072589 2667769941 1165584328 7910620065 4394330798 3081656781 9532745722 1220660395 6726593046 6339949932 4540890042 6080613659 5850529251 9825549915 0003838771 5930902111 3243761996 1612284069 0978886756 2380038387 7159309021 1132437619 9616122840 6909788867 5623800383 8771593090 2111324376 1996161228 4069097888 6756238003 8387715930 9021113243 761996161
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000002 0921288107 0647153218 4960243529 5674566916 3742480557 8791434503 1494660703 0351166660 6361161033 3444432929 4546173299 5498928163 6943201522 9459620954 6370526816 0174177523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 952380952
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0033220948 9790819676 1881600386 7010144869 9617284846 2544772788 2469670160 6626644574 6596112240 5822482653 4229525114 5021208127 8969815291 6432412252 5555673426 3241108334 8204158790 1701323251 4177693761 8147448015 1228733459 3572778827 9773156899 8109640831 7580340264 6502835538 7523629489 6030245746 6918714555 7655954631 3799621928 1663516068 0529300567 107750473
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000052754 6176658442 3525809730 8955042001 1593538592 6714254963 7673025007 1744556832 9605851687 4304480580 3336907823 7465564635 1959699909 3111572971 3519006232 6229657174 6615234521 5759849906 1913696060 0375234521 5759849906 1913696060 0375234521 5759849906 1913696060 0375234521 5759849906 1913696060 0375234521 5759849906 1913696060 0375234521 5759849906 191369606
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000083 7786553918 6585442850 1648187820 8912266237 8246372284 2329623008 8009653180 7581914097 4283811659 2903325353 8027766542 8694508812 0719935846 2383033535 4100202330 7980472014 8975791433 8919925512 1042830540 0372439478 5847299813 7802607076 3500931098 6964618249 5344506517 6908752327 7467411545 6238361266 2942271880 8193668528 8640595903 1657355679 702048417
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1330547517 7946606677 4433671206 9791307613 9584823074 0602827379 0927204979 6911412903 1284851480 5502074930 0258546131 2636477489 7806511217 8911052684 6832241947 9530899855 0003696857 6709796672 8280961182 9944547134 9353049907 5785582255 0831792975 9704251386 3216266173 7523105360 4436229205 1756007393 7153419593 3456561922 3659889094 2698706099 815157116
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0002113251 2502808408 6972384928 0674516249 3039311234 5924891389 6868672639 6888452328 4567853518 2414023112 0356520729 4150858963 1470718011 2113881481 1080033006 6608223755 8064005871 5596330275 2293577981 6513761467 8899082568 8073394495 4128440366 9724770642 2018348623 8532110091 7431192660 5504587155 9633027522 9357798165 1376146788 9908256880 733944954
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000003356 5666488978 0189143751 6525952965 7001694497 9900995455 9049306889 8757899326 8313557683 5480992765 3072770284 2915299724 9833592834 4367510573 1357489597 0524740202 3953792009 3260473588 3424408014 5719489981 7850637522 7686703096 5391621129 3260473588 3424408014 5719489981 7850637522 7686703096 5391621129 3260473588 3424408014 5719489981 785063752
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000005 3316602972 7274716354 1979233991 1122441751 2604602232 2033759333 0417122854 9162543633 0365798379 8282927886 8313231101 7693044896 8277924812 4546879600 1122077634 3468066927 4882314647 3779385171 7902350813 7432188065 0994575045 2079566003 6166365280 2893309222 4231464737 7938517179 0235081374 3218806509 9457504520 7956600361 6636528028 933092224
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0084693950 2877365653 5551065038 1244813530 4515534204 7667029569 5832590790 4883840248 8447230984 5717631480 4205895057 2331157641 1574743820 2406820608 7435893741 9943807083 6937910003 5906642728 9048473967 6840215439 8563734290 8438061041 2926391382 4057450628 3662477558 3482944344 7037701974 8653500897 6660682226 2118491921 0053859964 093357271
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000134544 1149668998 6995513563 5685856288 4462360506 3873227908 6495768279 7013961637 1297155387 2610523267 6922474676 7825421617 9853824263 2666639854 7709666432 2689222675 3529196195 0003565062 3885918003 5650623885 9180035650 6238859180 0356506238 8591800356 5062388591 8003565062 3885918003 5650623885 9180035650 6238859180 0356506238 859180036
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000213 7465444146 7136862253 0524940928 5035563566 0009438246 6739891330 2793981826 4875887303 8506292399 4426204625 2564247961 8483250960 4578905444 1303405657 6807160789 8644190807 9728005663 7168141592 9203539823 0088495575 2212389380 5309734513 2743362831 8584070796 4601769911 5044247787 6106194690 2654867256 6371681415 9292035398 230088496
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 3395902920 0503148898 9046739097 7141778821 5530138017 8822377612 9394948610 8455971532 7287890489 1548262370 7930260246 3225147379 3818171055 3980559365 0519215594 6468144724 9392045900 0351493848 8576449912 1265377855 8875219683 6555360281 1950790861 1599297012 3022847100 1757469244 2882249560 6326889279 4376098418 2776801405 975395431
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0005395514 8663417219 4721864138 3500156850 6791306522 9519720580 4167890333 7090991826 3235537339 4438508792 4299650451 8817661735 0334890162 1794666494 3210179221 4264685163 9748423243 6080139616 0558464223 3856893542 7574171029 6684118673 6474694589 8778359511 3438045375 2181500872 6003490401 3961605584 6422338568 9354275741 710296684
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000008572 9775207315 2631058934 2149727805 5471449341 1096054904 3815010609 8057963641 7788101110 2821527252 7918540588 4476346676 3928573704 1606345494 3535416104 5237500629 7371440822 2352510003 4662045060 6585788561 5251299826 6897746967 0710571923 7435008665 5112651646 4471403812 8249566724 4367417677 6429809358 7521663778 162911612
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000013 6223284804 4637709111 2014927071 7901568402 8084550564 3506454419 6126896922 6060072246 0327237235 7311143539 8816778938 4709911978 0442015452 9438677962 3884618348 6770625325 5887910218 9934595524 9569707401 0327022375 2151462994 8364888123 9242685025 8175559380 3786574870 9122203098 1067125645 4388984509 4664371772 805507745
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0216466949 6653496092 9739707822 5742497945 2773516920 0078980512 8691828313 4688111868 6494222563 6484369981 2558530047 3524831340 4697263072 7607805461 5963013916 6101370894 5344740856 9907287521 3675213675 2136752136 7521367521 3675213675 2136752136 7521367521 3675213675 2136752136 7521367521 3675213675 2136752136 752136752
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000343995 0167856829 1053032880 3428344614 7313745351 1665999873 9559347285 7409283654 8540846506 4848470917 3216561602 9275414672 5546344037 7522989653 5009408117 1917607611 4724690454 0613638745 5348047538 2003395585 7385398981 3242784380 3056027164 6859083191 8505942275 0424448217 3174872665 5348047538 2003395585 738539898
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000546 6794330840 2626455949 7241738401 2899204845 9004037164 0492682510 7623646726 6717276380 6875401684 6416255122 5204028922 0698406196 6630316467 8742989656 3400362821 9444378664 8969771152 0300033726 8128161888 7015177065 7672849915 6829679595 2782462057 3355817875 2107925801 0118043844 8566610455 3119730185 497470489
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 8688265428 9803035144 7321579499 8243616022 7098665042 6486598755 7222970378 5833603604 4053540001 8733466905 9635634141 7776547148 7427033239 9744039109 8591883189 6908892506 6620674235 9615512013 4003350083 7520938023 4505862646 5661641541 0385259631 4907872696 8174204355 1088777219 4304857621 4405360134 003350084
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0013808704 0561763492 7957540054 4729321633 9848460307 5686156404 5448684494 6848965487 7918935010 6652070228 9327394272 4329883790 0773361567 6386415151 6767678813 3490727743 9639993457 6209492075 0003327787 0216306156 4059900166 3893510815 3078202995 0083194675 5407653910 1497504159 7337770382 6955074875 207986689
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000021947 8509428416 1564833009 1179854075 5094873189 1434016211 5679509842 9918826388 7819184872 5723234100 3837053260 0535067134 3017163081 7908120953 4047146776 7717627576 5266045979 6840467847 2385459834 7107438016 5289256198 3471074380 1652892561 9834710743 8016528925 6198347107 4380165289 2561983471 074380165
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000034 8859108582 4414441310 9241809685 9525339634 2316762114 4380915871 1461677215 0153380904 6907225107 7326296005 8386400566 4794812173 8311551824 4382524857 2740805687 1823936834 9890544454 6406024835 9934318555 0082101806 2397372742 2003284072 2495894909 6880131362 8899835796 3875205254 5155993431 855500821
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0554532325 2899362857 4443140651 4812269968 9063545600 3332927837 5609214124 4344387364 3613876672 3409048520 1110753195 2887073515 4684846290 5346372483 2315299577 5882390283 9347688982 8982084878 6300032626 4274061990 2120717781 4029363784 6655791190 8646003262 6427406199 0212071778 1402936378 466557912
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000881499 6833782325 9166635060 5364226700 7868773947 3659755163 0844080254 8026666086 1052927861 0052381839 2034439913 6692693098 3856912313 1957153520 4725340106 6867075281 0218528708 8402375411 4245710003 2414910858 9951377633 7115072933 5494327390 5996758508 9141004862 2366288492 7066450567 260940032
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001401 3147945748 6507969890 7113712664 4106550811 0158449130 9130385307 6127877495 5880888003 8395883593 1588754265 8337750445 4536656270 2666454270 4595982054 3531741025 3098627587 0231353985 5171631827 5607085346 2157809983 8969404186 7954911433 1723027375 2012882447 6650563607 0853462157 809983897
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000002 2277542078 2333750910 2134573095 8453734830 2173205496 0903603121 3466305054 5273861200 5129840239 9010602537 9816519022 1481644194 6722191022 3410058508 4307285020 6123966074 1927458997 3120157653 4141829120 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000000000
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0035417395 9908320748 6648862596 3367970961 5742723697 1320991419 9067513998 4968638889 0310222321 5260748175 7360598599 3992816604 4430400939 6221637612 8909868497 6480324270 3882237613 6682226005 8082896389 6979332273 4499205087 4403815580 2861685214 6263910969 7933227344 9920508744 038155803
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000056309 7430097526 0686514084 2289942379 1085755888 0391686328 2227395807 4061409854 9414230975 8995005398 0474893691 0433419226 9047574653 2474498350 8587890824 1011579460 2617989074 1459647697 0809564971 4480126382 3064770932 0695102685 6240126382 3064770932 0695102685 6240126382 306477093
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000089 5298394352 8640538934 7951446238 0558260052 6898054473 9611051658 5145069695 2921266457 0367205873 5747919893 7681141235 4678322423 2477378605 8551801908 6760591095 2548715069 5985793298 5676665506 1272881431 7111459968 6028257456 8288854003 1397174254 3171114599 6860282574 568288854
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1423538414 2346942959 8788505638 0746123211 6157588919 0637459569 1893260304 2643396809 3182711788 1846293728 4800756036 1110589931 6907552175 1561429011 7087716684 0004426911 6988063526 5677069668 4615365555 0003120124 8049921996 8798751950 0780031201 2480499219 9687987519 500780031
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0002263536 4304482208 8366693785 0890067013 9005164687 1246384152 7661404229 0977573748 4769563084 0464529548 6014405667 2723031040 3627008349 6292095196 2704008003 4590865953 8614547221 6323222329 6929831812 2685400310 0775193798 4496124031 0077519379 8449612403 1007751937 984496124
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000003599 3368200655 8390130089 3470136439 3811176163 2621149874 3384509351 5630645386 9159140507 0920430499 9909964355 3907935543 9655690404 6484088508 8483067082 0640520452 4874961806 9841672674 0281096435 1775980482 4653312788 9060092449 9229583975 3466872110 9399075500 770416025
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000005 7236621040 6753801395 2293689723 8570495444 6063267690 7036953253 4619649099 8005719527 5690112978 6082552078 0806091884 0258098727 2835940394 3454641373 4425961893 4729908361 9301261244 8770456187 6543467347 8300030627 8713629402 7565084226 6462480857 5803981623 277182236
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0091021037 5392658099 9965534926 9763863704 1712704569 3808937912 8755206406 9702764651 0843289239 0259511492 2544575654 1863472848 6305417032 7610534869 5692128226 7675807927 3311181062 0222945772 6405803066 4916480779 2998477929 9847792998 4779299847 7929984779 299847793
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000144752 3671123694 6969990575 3634456540 8862705590 0806856211 0953324467 4357143035 1001813109 9062996669 4109801600 4957347780 1187298418 1313743107 7986250476 8411166126 0109084286 7358514917 6947046566 0869833895 0003025718 6081694402 4205748865 3555219364 599092284
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000230 2106818917 9240775130 8759464362 7633704056 2888260648 3140668023 9951278367 6299578221 8722359436 5076226355 1468936678 2139151408 7323544984 5275305651 8861761878 6389009575 2506042654 7021863453 3057046520 7930110075 1879699248 1203007518 7969924812 030075188
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 3661347765 8126625089 2963408176 5052021317 0641125723 9608462886 0650684843 3231065397 9492855890 0455068916 0566162852 9530935810 3122439838 3610422907 6436575358 2045916152 7474837712 0846085594 4907582507 4710370160 1195814648 7294469357 2496263079 222720478
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0005823338 2593252213 2178050567 7743405176 2522814652 2658658096 0969450955 5071219216 0504679276 2922804020 4572763775 2121857226 8816615541 7577736399 2278480995 9337625480 9062420603 6954165898 9901083322 9063848410 4300029717 6820208023 7741456166 419019316
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000009262 2904544186 0903883788 5544747560 8475457262 6551844672 1386664556 5567061578 8062675684 8518634084 7403669057 6560538871 2878968584 3653837085 0143109997 5427200524 5462922325 7105749255 3028545770 3428443060 5145510002 9542097488 9217134416 543574594
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000014 7326182382 1767593372 8630869413 4477387100 3038298029 5871668186 5604437945 8181541101 7648040212 1007206540 8564655470 9330088838 1359538673 0256791507 7082794043 4775162169 5495002859 9021204876 3266304977 2733470802 4011747430 2496328928 046989721
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0234345413 6111237912 0278417803 3764856293 4080161356 9825374846 2132500139 5412605265 9627488035 9344597740 5064484742 4833994255 1796101245 7490931094 9296334908 5982089075 4743666211 8483932123 8441540096 2091051663 9216004671 5328467153 284671533
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000372775 8683857644 0516437076 7960821890 4173805307 4706137316 1559502496 5935228179 0587738355 4243601856 4928810849 4597413726 7107066076 8727735936 8360564271 4946642045 9023510969 7651463555 3087564481 7785851658 3797649841 9738751814 223512337
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000592 9987262748 4369444906 5417893232 3982627998 5150874524 9312125823 3077124091 6949414029 9610675536 3442969005 8880871845 4434716925 8919834712 9823493053 4673819881 9500761209 9467086541 7364680708 6359490797 7896072913 6880115440 115440115
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 9433529235 2015306801 3090007690 1235052995 5878816771 4106796678 7846249519 6454395280 1670313223 8661385314 2521215366 8611588542 5070717808 7944832552 5074384980 6099085285 5517774900 3694080284 0311844194 2562254183 1372988051 649928264
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0015007520 4038470000 2065902962 3053148831 8126670551 3031058344 7306428156 1545058404 5883627294 3042161450 6454407502 6777796923 7412323629 8150750466 8510075540 8421913979 9007808020 7830127872 0866641610 0505485628 9087513775 000285307
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000023875 7941630564 4712506548 5993214768 4110086345 6600589553 2474466510 3828977216 3213990179 2528958990 4724027324 0496502095 0775434981 8201426782 5866951632 2363242579 7444776109 8951918115 4912516347 8437343195 2927433411 985872794
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000037 9857486825 4960840207 8798590051 1407159629 6162264407 6390170942 4876049487 0239922963 2188960036 3064215377 7454555788 9467524246 3462526388 4495878591 5954338134 8800448106 4202563841 5151962367 3347922881 6985757609 151932390
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0604362318 4499898425 1498161459 4025442190 5804160550 8331356255 1432790169 0576157319 2278758984 9441598942 3906233447 5273228554 7055472496 4026047860 6932761011 4729423882 6534686379 2732325927 2223065755 5638132739 736075669
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000961585 1225770549 6839761649 3624636296 3054590497 0103074492 6302195993 0611728424 3650978533 1241622002 8918817616 6205567223 3405970548 1504826939 1903471260 4768317052 6713125202 5404113022 8978733714 6638036236 861214665
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001530 0006277675 4155304541 6978482916 4481575903 2152152908 5408396503 8047708425 5872838283 7353675989 9607733242 1217574239 5542528889 6155870093 5709281084 9237434415 1495485849 1433486266 9346302917 2225552226 311965583
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000002 4344947919 8984266978 3714519141 9205718546 1544125135 9935071898 2162609429 5720792974 1298167173 2717499973 8870505646 4037928084 5890490223 0895771347 1166455592 1949193368 9318231268 6310838475 9245626754 549342736
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0038738188 7339947530 4456939426 8924936397 1379137428 1999347988 2056664783 1801146940 8093149346 9423802880 6168847443 8269251024 1126519847 2685650335 7512418502 7255776494 2496034496 9432264159 9891255043 944138872
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000061642 8694943130 6951048532 2612925028 3951182299 9948873178 0265253606 8255133503 1265484105 5910963060 9549134165 9733693389 3513301463 5059400467 3282677332 7988801006 4159518928 5637579526 4886921091 245916021
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000098 0932935453 6006502292 6982849984 3599752980 4632754461 9581947204 1113771534 0196293296 8236597527 1070392434 9387429681 9634073812 6952507007 9621202067 8099138759 5933848417 6382897901 3143364153 400017924
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1561020401 4667825732 1235490117 0061501793 1370611260 5418908275 4967451227 2508987137 7268751141 4358496905 2082048120 2172351551 7407965447 0496476817 1097995282 6252530626 2538179100 1420106147 111591500
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0002484222 5342000233 8094035969 2385536800 7059641606 3228167325 7066160282 1141324369 3295154136 0446796555 6118320102 3507779248 3218038770 2470094795 8582085615 0430610738 6717939953 2095682789 057738930
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000003953 5290530926 9408555528 5384944672 7188805197 3227460581 3781605958 6866222241 1859022461 5890056851 1272220343 8745021726 1201596665 1724091259 0075607271 0989737367 1568715333 0096472702 115563184
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000006 2920441131 8231685541 7601600687 5110043909 8917285244 0381455347 3849801435 2374995820 0769830868 6990582508 4463468836 7019686578 2782770452 8084922453 8561553919 3370528484 0040886604 098559483
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0100140749 6375436637 7023945057 4746542378 9995212080 9273674474 7721656857 0266501182 2510287308 0985582898 0266527856 3467087606 8553628377 5978119834 3096511175 9075525512 1763911836 434105607
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000159383 0170315281 0191657558 8089530896 3593222340 1740068691 8841577491 1763872255 2325130696 4357016738 5099735635 4002721861 3790055393 8120190142 8952902095 0542171191 1962523587 406424427
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000253 6794268465 2106783480 0570010212 8358262885 6488834911 9467165365 6905312014 1934262848 5437648501 4128231945 2413900279 9194968328 6895496919 6313585690 9685464245 2841837344 732060212
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 4037758367 4920908586 4494015204 8121079991 9012148678 7155042666 3297857089 6038657396 2634818831 9450315937 4525038509 6783113967 7949492962 5872936058 3689798450 2421688438 114371966
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0006426983 0470404509 6051323979 6998716916 9754660475 7235337082 1431464014 6999335011 7210382769 8458281589 5594328081 9952240547 7696958675 4736965149 4517686840 8967373409 384548413
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000010230 2350483650 3185065994 2060396413 2341251691 2179856711 5871230932 6813138709 8359029345 4436876708 7129563790 1825118315 4562182237 4386433305 6587412451 8466165826 811746216
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000016 2845431653 3598924038 4686260543 9448561644 1773651725 8179834973 4897687700 4405480083 5880353297 2141766291 5543596654 4043216011 4693773641 3387899450 9092646782 786033755
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0259225033 6229790083 4120641976 0200171875 9625019560 7599772639 2138507198 1534656162 9457982279 5926099810 3922628380 1010747009 8444995336 9964547158 1577786205 630279526
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000412657 3412299893 9730868854 6538417510 6162343503 8888650588 6373290196 5030174306 8956399525 6567405806 5422720939 4745335232 2488779358 8634569672 0287023700 220876592
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000656 9213462405 2788797287 3166645773 3485522499 9234669650 0669555663 9899942108 5057062495 2852321176 6584854530 7868759787 1339104457 8343369837 0163543932 811045189
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000001 0457990013 9269985478 0403354881 0002818608 4666284753 8921015724 2490219631 8033526224 4592445451 8305449013 0848963466 0199692667 1720203347 1131823719 626035952
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0016649224 5515102478 7535543518 2824613725 7047648497 2761088738 1417757213 5761243885 9443575888 1899804130 5636527588 2794864729 1999722351 1963030339 085227367
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000026506 3927766652 5922615593 2584233571 3675319585 4833652933 3252204966 0185331192 7505916629 6376450669 5325843197 2138372544 8906120303 3109232074 139840531
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000042 2005363366 4382433039 4117637197 5771586911 7751576940 6404485587 1399652294 9026733770 8516357068 5566847572 3584813626 4991583538 8759758579 618655271
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0671886521 9452861772 8489724316 8370773360 5148853187 1979889257 2496850811 7226593117 0885490249 1876257967 4017968244 4749575775 8090049977 251486331
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0001069755 1869601325 0846045818 1044573686 6430327574 3279157209 2360225096 1271589170 5345272760 9744715589 1848179404 0298813033 2548009225 226938474
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001703 2691600417 6031066931 9042949411 3558827922 2961771637 3981891944 2699846798 2505758027 4703359673 8345705164 5760631678 9245684235 882115291
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000002 7120174165 2225011405 4809084628 2990995001 6469433434 2095591046 3733249113 6414326743 9331939797 9000595052 4899237776 6950731067 651217875
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0043182906 9940816664 7228405297 5987392175 3946609954 3222637129 2897621096 2967511352 5958888010 2933993709 8543868508 8844843142 351386740
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000068760 8737538414 4346799227 8352160476 5683595178 2033770115 1119927256 4491332661 5343135497 0670506181 5807333237 9233906122 381770563
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000109 4916278842 5313722119 4872864706 6704543722 4056265960 3457840667 9180351312 0326080443 2383404600 5242661904 1093241278 470621773
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1743533746 2852770673 7959778526 0584697651 5494002558 1142628704 1884373753 2426564431 5452851195 5303473213 8192428143 543552743
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0002776448 5313626849 9629110861 5203127537 4673667135 9079272806 1353324986 1803116195 3835353084 6966930595 9248700273 739588408
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000004421 3880028296 2322141999 8642820528 1868738277 9360983718 2966960771 9450126988 3346979266 6319316007 2622973195 253352089
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000007 0410474361 5915613596 3571109504 5785334694 1110766659 1828068369 8858266181 1265578589 2564980256 5817879304 558537108
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0112130939 8001168762 4694377888 2752027800 2223136492 8079961432 5685182919 1178547563 1955860776 0258842755 746629512
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000178576 0100510107 5576052184 6229999512 7986815754 6561420723 7123553452 7510968669 4757883712 3067208972 342252937
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000284 4003600534 9412385629 4676724216 5651230794 6631379205 9988279083 5104160821 3168437526 7075996979 283582298
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 4529460279 7932781060 6572879846 1630910590 8168167812 2222121724 1099866695 7154229134 1046758549 849630009
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0007213930 7823878280 3971797173 1249772758 4026021787 5442311154 6909536374 1737946437 7870093256 686954096
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000011489 6448310596 5587899437 0483705453 3511251479 1708572200 9360003694 8773614097 5875722857 766180544
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000018 2999655243 3928391532 9058641016 4496507364 4444637308 1865532301 0041451912 9418103080 946671401
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0291476061 0520751932 9997560520 1997598052 8597976452 8244291134 8378479582 6174946619 325313478
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000464263 3323394133 5625732897 2965161528 5071422648 6582670321 9875482482 6647765719 249679014
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000739 4940245460 0398425579 4771719648 1054295175 3364518690 1074972996 9578302107 456954920
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000001 1779142277 2723577782 0155596246 4138383920 7876485199 8934510503 2531252483 892011703
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0018762957 8205785659 0818520878 7268070375 8081181278 0897747846 2533505941 803420439
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000029888 0441924265 8631297215 1367652652 2141049536 9397726180 4366395913 995252830
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000047 6104378334 8041472077 4120110408 1852651528 8553232021 4623457141 125924231
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0758429581 4349826415 1033822879 0554924968 6567000898 8521520652 865337945
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0001208194 0376883562 0798070020 1327418358 1180084051 1592619669 977868796
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000001924 7147579764 9970504808 1797380660 3854839760 3907284138 523117838
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000003 0662266674 0991347606 3625228778 5309665396 6172019215 034083964
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0048848390 3957661135 0456582711 0788222071 8912739061 828041382
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000077822 3445704895 5103235438 9514979221 1014567947 847268168
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000123 9842006134 7577564685 0194949798 2724228972 286602350
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1975314703 0788829208 2633169254 7689373181 801543118
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0003147125 7322281123 0221812168 9206138751 600550691
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000005014 1771413370 9778814942 0436332646 778103301
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000007 9890100434 4832033738 3084984813 736482552
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0127289890 4310344707 4651753380 192252235
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000202816 1084578563 8044439921 722945241
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000323 1606718806 2173116197 744705907
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 5149226804 7851388329 627778974
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0008204893 3588272140 253088292
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000013074 0818638917 162687828
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000020 8332362585 479417649
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0331978148 542303040
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000529016 746312103
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000843 017219904
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000001 343416007
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 002140876
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000003412
0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000000005


Written at normal size, this 253 × 709 array would fill a sheet of paper roughly two meters wide by one meter high. I’m sure that Shanks never actually created such an unwieldy artifact. Instead he must have broken the process up into small sections, with some sort of hierarchical system of subsums copied over from one sheet to another. But I was curious: If he had decided to tackle the summation all in one go, could he have found paper in large enough sizes? The British Association of Paper Historians suggests not: The largest sheets in their list of Old English Paper Sizes are Imperial, Elephant, Double Elephant, Atlas, Columbier, and finally Antiquarian, which is 31 by 53 inches. But another source, paper-sizes.com, includes a sheet called Emperor, which is 4 feet by 6 feet—just about right. Of course even without the Emperor, Shanks could have pasted together smaller sheets. (Scotch tape was still in the future.) I like to imagine him taking over the family dining table with this huge army of numbers in tidy rows and columns.

Shanks’s 707 decimal places of π were accepted as correct for more than 70 years; his value was enshrined in carved wooden numerals on the ceiling of the Palais de la Decouverte in Paris. His errors were not discovered until the 1940s, when D. F. Ferguson, working with a mechanical desk calculator, was making rapid progress on a new computation of the constant. His value of π differed from Shanks’s starting just before decimal place 530. Many subsequent computations confirmed that Ferguson was right and Shanks was wrong.

But where and how and why did he go wrong? If we had all of his working papers, it would be easy to track down the mistakes. In 1853 Shanks published a small book that included the individual terms he summed to calculate arctan 1/5 and arctan 1/239. Unfortunately, the terms are given only up to the 530th decimal place—exactly where the trouble begins. For all the remaining digits—up to 607 in the book, then on to 707 in an 1873 publication—we have only the sums of the terms, that is, the computed values of arctan 1/5 and arctan 1/239, along with the resulting value of π.

In the absence of the working papers, we can never identify Shanks’s errors with complete certainty, but there’s hope of finding some plausible candidates. The basic idea is to take the difference between the true arctan value and the value that Shanks reported, then subtract this difference from each of the individual terms. If we get lucky, an error will show up as some sort of distinctive pattern.

As far as I know, Engert was the first to identify a likely error in this way. Shown below are decimal places 520 through 580 in the true and the Shanks values for arctan 1/5, along with their difference. When we subtract this difference from the true value of term 248 in the arctan series, we get an adjusted value that immediately suggests a clerical blunder. If we omit the 0 at position 530 (marked in red), letting all the following digits slide left one place, the true value of term 248 is converted into the adjusted value. It’s just the kind of slip that might well happen when copying a number from one page to another. This anomaly popped up when I ran a program searching for shifts caused by omitted or repeated digits; then I discovered that Engert had already spotted it some time ago. There is also a single-digit substitution of 2 → 3 (flagged in green) that Engert apparently didn’t notice.

a5 true     0.197 ... 560518377574220877835853152464749330914587633823112490332030
a5 shanks   0.197 ... 560518377617781642423378303370181926488028277686119150985606
diff       -0.000 ... 000000000043560764587525150905432595573440643863006660653576

t248 true   0.000 ... 744466800804828973843058350100603621730382293762575452716297


The omission at decimal place 530 in term 248 “uncorrects” the value of arctan 1/5 through place 568; then something else goes wrong. In my American Scientist column I suggest the following explanation of this second error. Starting with the “uncorrected” value of arctan 1/5—that is, the true value with the error in term 248 already incorporated—we subtract the Shanks value of arctan 1/5 to get a second difference. Then, subtracting this difference from term 72 in the series, we observe this intriguing pattern:

a5 248ucx   0.197 ... 018192648802827768629156477871020728799805452914758511130462
a5 shanks   0.197 ... 018192648802827768611915098560675901213598556363034321005664
diff        0.000 ... 000000000000000000017241379310344827586206896551724190124797

t72 true    0.000 ... 413793103448275862068965517241379310344827586206896551724137


We can convert the true term 72 into the adjusted value by excising the five digits marked in red and letting all the rest of the digits shift to the left to fill up the void. Again, one can imagine the omission as a careless error, probably made while transcribing intermediate results for use in a later stage of the computation.

This much of the story is told in my American Scientist column. But something about those numbers bothered me, and I kept poking at them even after the magazine went to press. Here they are again with different highlighting:

t72 true    0.000 ... 413793103448275862068965517241379310344827586206896551724137
diff        0.000 ... 000000000000000000017241379310344827586206896551724190124797


The very same motif—the 28-digit sequence 1724137931034482758620689655—appears in all three numbers, shifted left or right by a few places. The presence of these digits in t72 true is easy to explain: This sequence is the repeating unit, or repetend, in the decimal representation of term 72, which is equal to $$(1/5)^{145}/145$$. If you were to compute still more digits of the term, you would find that the same sequence repeats endlessly. But seeing the same sequence turn up in the other two numbers seems a bit surprising.

Try an experiment (or consult my IPython notebook, which includes code for the experiment). Take the sequence 1724137931034482758620689655 and create a cyclic permutation: for any k between 1 and 27, lop off k digits from the front of the sequence and paste them onto the back. Now subtract the permuted sequence from the original one. Notice anything?

Here’s an example, with k = 7:

 1724137931034482758620689655
–9310344827586206896551724137
–7586206896551724137931034482


The digits in the result form another cyclic permutation of the same sequence. That can’t be a coincidence!

And indeed it’s not a coincidence. The sequence 1724137931034482758620689655 is a cyclic permutation of 0344827586206896551724137931, which is the repetend of 1/29. It’s also not a coincidence that 145, the denominator in our term, is equal to 5 × 29. And the secret behind all of this hocus-pocus is that 29 is a “full-repetend prime” in base 10: a prime p whose reciprocal 1/p has a repetend with p–1 digits, the most possible. For all such primes the repetend is a “cyclic number”: multiplying by any integer in 1, 2, 3 … p–1 yields a cyclic permutation of the same digits. And, as observed above, subtracting any two cyclic permutations also yields another cyclic permutation.

I was led to look at term 72 by this note at the end of the 1946 paper in which Ferguson announced the discovery or errors in Shanks’s work:

In calculations made since the preceding part of this Note was written, from the 569th decimal place onwards I have a further disagreement with Shanks’ figures, namely 6896551724137931034482758620 (recurring). These are precisely the figures which should occur at this exact spot (i.e. from the 569th d.p. onwards) in the term 145 which comes in the series for $$\tan^{-1}(1/5)$$. This surely cannot be a mere coincidence. It looks very much as if Shanks omitted the whole of this term from the 569th place onwards.

I am deeply puzzled by this statement. The sequence of digits mentioned does indeed appear exactly where Ferguson indicates in term 72. (In Ferguson’s numbering this is term 145; 2 × 72 + 1 = 145.) But truncating that term at position 569 does not have the desired effect of transforming the true value of arctan 1/5 into the value given by Shanks. To accomplish that, as noted above, what’s needed is not truncating the term but shifting the digits left by five places.

At a loss to explain the discrepancy between Ferguson’s observations and my own results, I went looking for other occurrences of the same cyclic number. It has to appear in at least one other place, namely term 14, which has the denominator 2 × 14 + 1 = 29. Here’s what I found:

t14         0.000 ... 4137931034482758620689655172413793103448275862068965517241379310
t14-diff    0.000 ... 4137931034482758620689654999999999999999999999999999999999478062
diff        0.000 ... 0000000000000000000000000172413793103448275862068965517241901247


Hence in term 14 we see the very phenomenon that Ferguson reported in term 72: truncating the term beyond decimal place 568 would recreate Shanks’s erroneous value, through decimal place 601. (The truncation could be expressed by setting the sigits to 5000… or to 4999…; it doesn’t matter which.)

With this discovery we have an alternative theory of how Shanks goofed in the neighborhood of decimal place 569. Instead of skipping five digits while transcribing term 72, he might have neglected the trailing digits of term 14 while summing up the series of terms. Which mechanism is more likely? If we could diagnose the further error beginning at place 602 in the arctan 1/5 series, we might learn the answer to this question. Within the range 569–601, the shift in term 72 and the zeroing of term 14 have identical effects, but in the later digits they would yield different values. Unfortunately, my attempts to find the source of those further errors in the arctan 1/5 computation have come up empty.

There’s also an error in Shanks’s value for arctan 1/239 that remains unaccounted for. And there’s the further mystery of Ferguson’s note. Why did he point to term 72 instead of term 14? To explain away his comment as a result of simple confusion or a memory lapse, we need to assume that he was mistaken both about which term he was looking at and which permutation of the cyclic number he was seeing. That’s hard to credit in someone who was an exceedingly meticulous calculator. I can offer up another hypothesis, though it’s no less farfetched: Maybe Ferguson derived the error term not from arctan 1/5 but from 4 × arctan 1/5, which is how the number enters into Machin’s formula. It turns out that 4 × 1724137931034482758620689655 is the very permutation that appears in term 72: 6896551724137931034482758620.

Even if we solve this mystery, there’s another intriguing conundrum in Ferguson’s observation. I don’t understand how he could have spotted the error at position 569—whatever its cause—without having discovered the site of the earlier problem at position 530. The distinctive repetend patterns appear only after the single-digit omission in term 248 has been “uncorrected.” Ferguson knew there was an error somewhere near decimal place 530, but his publications on the subject (or at least the three known to me) say nothing about the cause of that error.

In his 1853 book on π, Shanks ends his introductory remarks by declaring that he is going to “take leave of the subject for the present.” I’m going to follow his lead and declare that I’ve had enough of Shanks for the present. Maybe, like him, I’ll return to the topic after a suitable interval—say 20 years.

Posted in computing, mathematics | 6 Comments

## A Glitch in the Maptrix

We must always be on the lookout for glitches in the Matrix—anomalies that give us a fleeting glimpse into the algorithms and data structures of the computer simulation that we call Reality. But there’s also the Maptrix, the alternative reality supplied by online mapping services. I spend a fair amount of time exploring that digital terrain, and lately I’ve noticed a few glitches. Exhibit 1 is a tank farm in Bayonne, New Jersey, near where Kill Van Kull enters New York Bay:

I’ve seen a lot of oil tanks over the years, but never before have I encountered such ragged, faceted approximations to cylindrical form. These lumpy, polyhedral tanks suggest that in this little corner of industrial New Jersey, π has a value somewhat smaller than 3.14.

But the π peculiarity is ephemeral. The image above was captured from a laptop screen at the instant the landscape was first rendered in the Apple Maps program. Moments later most of the defects were magically healed, and the illusion of solid, circular reality reasserted itself:

Here’s another example, from a tank farm in Carteret, New Jersey, just across the Arthur Kill from Staten Island. This time we’re looking down on the tanks from directly overhead. The image at left was captured as soon as the scene appeared on screen; at right is the rounded-out version that emerged a few second later. The software, again, is Apple Maps.

It’s not just Apple’s version of reality that has such anomalies. Here’s a sample from another source:

The mangled petroleum tanks in this image are in Kearny, New Jersey, a few miles north of Bayonne along the Passaic River. In this case the picture was taken by Google’s eye in the sky, not by Apple’s. The distortion is different, but no less disturbing. Now it’s not just the geometry of the cylinders that has gone goofy but also the topology. Some of those tanks won’t hold oil (or any other fluid); they have holes in them. And notice how the spill-containment walls surrounding the tanks also look moth-eaten.

Finally, returning to Apple Maps, and scrolling just half a mile northwest from the Carteret tanks, we cross the Rahway River into Linden, New Jersey, where we come upon this alarming scene:

Toward the right side of the image we see more cylindrical tanks, some with faint, remnant traces of polyhedral approximation. But when your glance wanders to the upper left, you find that the world suddenly loses all depth. The tank farm over there, and the water treatment plant at the top of the frame, are merely painted on the landscape—trompe l’oeil structures that don’t trompe anyone.

This image offers another view of the same Linden landscape, looking obliquely to the west or northwest. The road that runs through the scene from foreground to background, crossing the New Jersey Turnpike at the very top of the frame, is Tremley Point Road. Suppose you were driving west along that road. Just beyond the row of lumpy trees that extends from the left edge of the image toward the road, you would cross a mysterious boundary, leaving behind the pop-up 3D world and entering flatland. What would happen to you there? Would you be pancaked like those tanks, reduced to a two-dimensional object painted on the pavement, with a painted shadow to accompany you?

Leaving behind tank farms but still poking around in the same general neighborhood of northern New Jersey, I was able to record four stages in the “construction” of the Bayonne Bridge, which crosses Kill Van Kull between Bayonne and Staten Island. These are images from Google Maps, the first three captured at intervals of about a second, the last after a delay of a few seconds more:

In calling attention to these oddities in online map imagery, my aim is not to mock or belittle. To me, these maps are one of the marvels of the age. A century ago, it was a huge novelty and liberation to soar above the earth’s surface for the first time, and see the landscape spread out below as if it were a map. It’s no less remarkable that we have now transformed the experience of looking at a map into something like flying an airplane.

The first “satellite view” maps were just that: montages of images made from hundreds of miles above the earth’s surface, looking straight down. They portrayed the territory as an array of pixels, assigning an RGB value to every (lat, lon) pair on the surface of a sphere.

The next step was to add a digital elevation model, giving each point on the surface a z value as well as a color. This scheme allows us to gaze obliquely across the landscape and see a realistic rendering of mountains, river valleys, and other natural landforms. It works well as long as you don’t try to get too close: the model is well-suited to forests, but not to trees. And it doesn’t work well at all for manmade artifacts.

In representing engineered structures, one weakness of elevation maps is that any reasonable scheme for interpolating between sample points will tend to round over corners and sharp edges, so that buildings become lumpish mounds. Symmetries are also lost: the sampling fails to preserve the rectilinearity or circularity of the various objects we strew around the inhabited patches of the world. And the biggest problem with an elevation map is that it’s a mapping in the mathematical as well as the cartographic sense. The surface of the planet is defined by a smooth, single-valued function, assigning a unique elevation z to every (lat, lon) point on the sphere. Any line radiating from the center of the earth must cross that surface in exactly one point. As a result, there can be no vertical cliffs and no overhangs. Also no bridges. The surface defined by the elevation model can go under a bridge or over it, but not both.

The latest mapping programs are apparently addressing these issues by building explicit three-dimensional models of selected landscape features. I see evidence of several different techniques. The Bayonne Bridge model that assembles itself in the four frames above is clearly based on a triangulated mesh: all the surfaces making up of the envelope of the structure are decomposed into elementary triangles. The cylindrical tanks in the Apple Maps images seem to grow their circular form through an Archimedean process, in which a circle is defined as the limit of an n-gon as n goes to infinity. Elsewhere, I think we may be seeing some kind of spline curves or patches.

Having constructed the model and embedded it in the landscape, the next trick is to project the pixel pattern from a photograph onto the model surface, as a texture. This process too has a certain comical potential:

What we’re seeing here is apparently a tugboat nudging a barge equipped with a crane. The matching of surface pattern to three-dimensional form has gone badly awry, which gives the whole scene a toylike quality. I find it charming. In future years, when the Maptrix has become a hyperrealistic, real-time virtual world with day and night, weather and seasons—maybe with inhabitants who wave back at you—we’ll wax nostalgic over such quaint foibles.

Posted in computing, modern life, photography | 9 Comments

## Cosmic eggshells

In the creation myths of certain cultures, the world began as an egg, which broke open to form the earth and the sky. What if some intrepid explorer mounted an expedition to the ends of the earth and came back with fragments of the original eggshell, or maybe feathers from the bird that laid it? That’s what happened last week. Thanks to a group called BICEP2, which took a telescope to the South Pole and pointed it at a single patch of sky through three long Antarctic winters, we now have bits of eggshell providing strong evidence for our modern creation myth: the inflationary universe.

In referring to cosmic inflation as a creation myth, I don’t mean to be dismissive—not in the least. It’s just that when you listen to this narrative of how the world began, the story is every bit as fantastical as any tale of hatching eggs or turtles-upon-turtles. Last Tuesday I heard Alan Guth, who dreamed up cosmic inflation in 1981, tell the story to an overflow crowd at Harvard. The video of the colloquium is on the net. Here’s my paraphrase of a few highlights:

Once upon a time, the entire universe we know today was packed into a tiny dot, much smaller than an atom—only a millionth of a billionth the size of a proton. With so much matter and energy confined in such a small space, gravity was a very powerful force, which you might think would hold everything together forever. But one day something odd happened. Inside the microdot universe, gravity suddenly changed its sign; it became a repulsive force instead of an attractive one. That repulsion is what put the bang in the big bang. The universe doubled in size in 10–37 second. Then it doubled again, and again. After about 100 doublings, the dot had expanded by a factor of 1028, and our universe had a diameter of 1 centimeter—the size of a marble. It was all over in a trillionth of a trillionth of a trillionth of a second, or thereabouts. Afterwards, gravity resumed its familiar, convivial, let’s-all-stick-together personality, and the rest of the history of the universe unfolded at a more stately pace over the next 13.8 billion years.

Three decades ago, when I first read about (and wrote about) the inflationary universe, physics itself was in a bubbly, inflationary mood. Theorists, it seemed, couldn’t dream up an idea too outlandish for experimentalists to confirm. Let’s have three quarks for Muster Mark, decreed Murray Gell-Mann; sure enough, dense little quarky pits were found inside the proton and the neutron. We need a fourth quark—and let this one have charm—said Sheldon Glashow and his friends; and there it was in the J/ψ particle, discovered in 1974. Give us supermassive analogs of the massless photon, said Steven Weinberg and Abdus Salam; nature obliged with the W± and the Z0. Now and then, the theorists had to scramble to catch up with experiments, as when Leon Lederman found evidence for a fifth quark. On the other hand, a few theoretical predictions were way out in front of experimental prowess: Finding the Higgs boson took 50 years.

Not every flight of fancy landed in Stockholm. Theorists asked experimenters to keep an eye out for decaying protons; they spent years looking and saw nothing. The magnetic monopole is still missing in action. Supersymmetry has not checked in yet. String theory remains… well, a theory. My own sentimental favorite was the preon or rishon model, a fantasy of recursive descent: If the proton is made of quarks, the quarks must be made of something still smaller. If only it were so.

In that giddy era, Guth’s inflation conjecture was far from the wildest or weirdest idea to come along, but it did seem to be among the proposals that would be most difficult to test. Yet here we are with solid observational evidence.

Guth’s talk on Tuesday was the first half of a double feature organized by the Harvard physics department. The second speaker was John Kovac, leader of the BICEP2 collaboration—the group that brought back the eggshells from the South Pole. Written on those eggshells was a distinctive pattern of polarization in the cosmic microwave background (CMB) radiation.

The peculiar distribution of the CMB was one of the puzzles that led to the inflationary hypothesis in the first place. The CMB is al­most perfectly uniform across the entire sky, suggesting that whatever emitted the radiation must have been at a uniform temperature every­where. In a non-inflationary cosmology, that widespread thermal equilibrium seems mysterious. Distant regions of a very young universe are causally dis­connected: They have not yet had time to communicate with one another, even by signals moving at the speed of light. Some sort of miracle is needed to make all of these isolated regions so nearly uniform in temperature. Inflation dispenses with the miracle by squeezing the early universe into a much smaller volume, where all the parts can become throughly mixed.

Inflation predicts a CMB that is nearly uniform, but not quite. Quantum fluctuations introduce variations in density and temperature, which become slightly warmer and cooler spots in the CMB radiation field. (Later in the history of the universe, the denser spots evolve into clusters of galaxies.) A series of satellite observatories has mapped the CMB with high precision, finding splotchy patterns with an angular size scale of about 1 degree, and temperature differences of 1 part in 100,000. These results are in good agreement with the predictions of inflation. This is encouraging news for the theory; on the other hand, inflation was invented to explain these very characteristics of the CMB, so there’s some circularity in the argument.

Inflation makes another prediction. Quantum fluctuations in the high-density plasma of the inflating universe should give rise to powerful gravitational waves. These are propagating disturbances in the gravitational field, just as light waves are disturbances in the electromagnetic field. We are too late to detect the gravitational waves directly, but they should have left a distinctive signature in the microwave background—a swirly pattern of polarization, or wave alignment.

If the inflationary scenario is correct, two modes of polarization should co-exist in the CMB; they are called the E-mode and the B-mode, after the E and B vector fields of electromagnetism. In the argot of vector calculus, the E field has divergence but no curl: Think of electric lines of force streaming radially away from a charged particle. The B field has curl but zero divergence, like the magnetic lines of force that wrap around a current-carrying conductor. E-mode polarization comes from a conventional process of photon-electron scattering. B-mode polarization is the mark of gravitational waves.

The search for CMB polarization takes us to a rough neighborhood of the electromagnetic spectrum: the millimeter band, in the no man’s land between radio and optical wavelengths. Nearly anything can emit and absorb at these wavelengths, so it’s a noisy place. But astronomical signals are very faint because moisture in the atmosphere soaks up millimeter waves and not much gets through. I gather that the best place on earth to do millimeter-wave astronomy is the South Pole, which has exceptionally dry air and dark skies. In the past decade half a dozen millimeter-wave telescopes have been set up at the Amundsen-Scott South Pole Station. E-mode polarization was first detected in 2002 by one of these instruments, called DASI. BICEP2 was installed in 2009 to look for B-mode signals.

The BICEP2 telescope was operated at the Amundsen-Scott South Pole Station, on a platform located some 800 meters from the geographic pole. The telescope is inside the conical shield on the roof of the building; the shield blocks stray radiation from nearby sources. The instrument at right is the South Pole Telescope, which also operates at millimeter wavelengths. Just visible in the background at left is the IceCube neutrino telescope. Photograph by Steffen Richter, who spent three “nights” at the station (each night being six months long). Reproduced courtesy BICEP2 collaboration.

The BICEP2 instrument is a curious hybrid of optical and radio technologies. The telescope is a refractor, with objective and eyepiece lenses—a design known to Galileo. But the lenses are not ground from optical glass; they are molded of high-density polyethylene (the material used to make plastic milk jugs). And the detector in the focal plane of the instrument is not the kind of imaging device seen in optical instruments; it is an array of tiny crossed antennas, etched into silicon wafers. The whole instrument is evacuated and cooled to liquid helium temperature. The detector is colder still: about 0.25 Kelvin.

The telescope was installed at the South Pole in December of 2009 and recorded data almost continuously until December of 2012. (It was shut down briefly every three days to replenish the helium and brush off any snow, and there were slightly longer interruptions in summer.) Most of the observing time was spent repeatedly scanning the same patch of sky, an area known as the Southern Hole, which is unusually free of “nearby” dust—that is, schmutz from our own galaxy. In a sense, the entire three-year observing campaign was an effort to take a single picture of that patch of sky. Here it is:

Polarization maps of the target area for BICEP2 reveal both E-mode and B-mode signals. E-mode polarization had been seen in earlier observations, but this is the first evidence of B-mode fluctuations, thought to result from gravitational waves in the inflationary era. Note that the intensity scale at right is six times larger for the E-mode map. Graphic reproduced courtesy BICEP2 collaboration.

The important features of these vector maps are the pinwheels in the B signal panel. They are interpreted as the telltale remnant of gravitational waves.

The BICEP2 group calculates that the B-mode signal is statistically significant at the 5σ level, which pretty much rules out the possibility of being fooled by mere random noise. More worrisome is the risk of some systematic error, such as a bias in the instrument, or polarization caused by non-cosmological sources, such as galactic dust. Kovac devoted much of his Harvard talk to these issues, and there is even more detailed discussion in two papers released on the BICEP web site. As far as I can tell, the checks for errors were very thorough and the evidence is very solid. In any case, we won’t have to wait another 30 years to see the result confirmed (or not). Another telescope, the Keck Array, is taking data now at the South Pole, and BICEP3 is being readied for installation.

In case it’s not obvious already, let me say that I’m really wowed by this story. I think it’s a big deal. The discovery of the CMB in 1964 transformed the big bang model from a plausible possibility to a prevailing theory, and the discovery of B-mode polarization will surely do the same for inflation.

To me it’s amazing that we can have such detailed knowledge of our most remote origins. A hundred years ago we knew nothing about the early universe. We didn’t even know that it had a beginning. We didn’t know how big it was or what it was made of. We had none of the conceptual tools that would prove essential to making sense of cosmic evolution: quantum field theory, general relativity, the nuclear chemistry that makes the stars shine. In one quick century it has all come together.

Or not quite all. We still don’t know what the universe is made of. Most of it is apparently “dark matter” and “dark energy” that we have yet to identify; all the stuff we can see is a minor constituent. And we still don’t have a satisfactory quantum field theory for gravity. Yet, somehow, we can reconstruct the dynamics of the exotic, singular event that created this place, and supply enough quantitative detail to detect the evidence 13.8 billion years later. It’s magnificent. It’s ridiculous.

Posted in physics, science | 3 Comments

## Net pests (not)

Last night I posted this note:

Since Sunday afternoon bit-player.org has been under some sort of mysterious DDoS attack, with a rotating suite of IP numbers repeatedly downloading the same PDF files several times a second. For the time being, I’ve taken most PDFs offline. If you urgently need something and you get a permission-denied error, please send me an email. Sorry for the inconvenience. Back soon, I hope.

Turns out it was not net pests. It was my 15 minutes of fame. A link to an old story of mine found its way to the front page of Hacker News, and I misinterpreted the resulting net traffic jam. A couple of hours later (and after a couple of messages from helpful HN readers), I realized there was nothing malicious going on, and I put the files back on line.

I’m an amateur in all things, but I’m especially inept as a sysadmin. Looking at the server logs this morning, I’m still unsure of exactly what I’m seeing, but I understand a little more than I did 12 hours ago.

What set me off in the first place was seeing long lists of requests like these, all from the same IP number:

16/Mar/2014:14:17:25 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 200 79640
16/Mar/2014:14:17:26 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 65886
16/Mar/2014:14:17:26 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 8781
16/Mar/2014:14:17:27 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 65891
16/Mar/2014:14:17:27 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 65892
16/Mar/2014:14:17:27 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 65893

Good grief, I thought: Somebody is downloading the same PDF file six times within three seconds. I failed to notice (or appreciate the significance of) the response codes near the end of each line. The “200” code on the first line is the normal HTTP “OK” signal, but the “206” on the next five lines signifies “partial content.” What’s going on here—if I now understand correctly—is not one person downloading the same file six times; it’s one person downloading a file in six pieces. (The size of the file in question is 270,570 bytes. The byte counts at the ends of the six lines above add up to 272,343. I can’t account for the discrepancy; I’m still an amateur.)

The file mentioned in the six requests above is not the one linked to by Hacker News. That’s another reason for my confusion: I was seeing wholesale downloading of hundreds of different files. Apparently, when some people find an item that interests them on a web site, they wget -r the whole site. And I guess I understand why: If you don’t grab it immediately, the skittish site owner is likely to panic and take it offline. (But wouldn’t it be polite to throttle the request rate?)

The story that started all this fuss is a bit of whimsy I wrote almost 30 years ago for Computer Language, a magazine long defunct. In the past 18 hours the PDF has been successfully downloaded almost 12,000 times, which may be greater than the circulation of Computer Language.

Posted in meta | 4 Comments

## Notes from the JMM

The annual Joint Mathematics Meetings are one of those vast smörgåsbord meals where you can’t possibly eat everything. I managed to sample 1 percent of the menu this year. By my count, the four days of meetings included 2,743 scheduled events (mostly talks, but also panel discussions, cocktail parties, poetry readings, film showings, and other diversions); I attended 28 of them. There are $\binom{2743}{28} \approx 5 \times 10^{66}$ ways to choose such a sampling, so it’s unlikely any of the other 6,424 participants experienced exactly the same meeting I did. Technicalities: I came up with the number 2,743 by scraping the online program and counting all <li> tags that do not have a <ul> as one of their children. Counting links to abstracts gives an estimate for the number of talks alone: 2,569. Also, I should note that the binomial calculation exag­gerates a little: Some of those 1066 sets of 28 events are forbidden because of conflicts between simul­taneous sessions.

As a journalist and a generalist, I try to maintain a balanced mathematical diet, but this year I pigged out on number theory. A third of the talks I heard were in that area. There’s been a a lot of buzz lately about progress toward a proof of the twin-primes conjecture, starting with a surprise announcement last spring by Yitang Zhang of the University of New Hampshire. Twin primes, such as $$5$$ and $$7$$ or $$101111$$ and $$101113$$, are primes as closely spaced as two odd primes can be. The twin-primes conjecture says there are infinitely many of these pairs. Zhang didn’t prove there are infinitely many primes $$p$$ and $$q$$ with $$q-p=2$$, but he came really close. He proved there are infinitely many primes $$p$$ and $$q$$ with $$q-p \le 70\,000\,000$$. A few months later James Maynard, a postdoc at the Université de Montreal, brought the constant down from $$70\,000\,000$$ to $$600$$. (Maynard was inspired by Zhang’s work but used different methods. Terry Tao got the same result at the same time.) Then a Polymath collaboration organized by Tao made further progress. At last report, the minimum gap for which we can say with certainty there are infinitely many primes is 270.

For more about the mathematics behind this work, I recommend two excellent articles by Erica Klarreich, published in the Simons Foundation’s Quanta magazine:

I’d like to add a few thoughts about the human side of Zhang’s story, as best I can piece that story together from published sources.

According to a profile by Virginia Stuart in UNH Magazine, Zhang was born in China in 1955 and was a boy when the Great Proletarian Cultural Revolution and the Red Guards movement convulsed the nation. With the rest of his family he was sent to the countryside for “re-education” through labor. The result of re-education was not much education, but he made up for lost time and in 1978 (at age 23) entered Peking University, earning bachelor’s and master’s degrees in mathematics.

In 1984 S. S. Chern took several American mathematicians to Beijing, where they spent a summer working with Chinese graduate students. Zhang was one of the students, and the following year he came to the U.S. to study at Purdue with T. T. Moh, who had been a member of the Chern delegation. Moh has recently written a short memoir on Zhang’s period at Purdue, from which I quote at length:

When he arrived, we had a cordial talk. Yitang expressed his desire to work in the field of Algebraic Geometry…. Yitang also mentioned that he wanted to study under my guidance…. I was surprised by Yitang’s next request of working on the Jacobian conjecture as his thesis topic. I felt it was odd to select such a difficult task.

The Jacobian conjecture is one of Steve Smale’s “problems for the 21st century,” which Smale describes as follows:

Suppose $$f : \mathbb{C}^n \to \mathbb{C}^n$$ is a polynomial map with the property that the derivative at each point is non-singular. Then must $$f$$ be one-to-one?

I can say nothing more about this question except that it remains open, and that Moh is the author of a long 1983 paper with computational results for all two-variable polynomials of degree less than 100. Moh continues:

Yitang spent all of his free time thinking of mathematics. After years, Yitang started to believe that he might have gotten a solution, one independent of my paper, to the Jacobian conjecture. As a gatekeeper of the palace of the Jacobian conjecture, I did my duty of examining every claim presented to me and denied the entrance of anybody (even if the claim has nothing to do with my work) if the proof was invalid. “Maybe the Jacobian conjecture is a problem for the future”, I thought.

By 1991 Zhang was approaching the seven-year limit for Ph.D. candidates at Purdue. He wrote a brief thesis on a specialization of the Jacobian conjecture, which earned him his degree. Then he took off. Moh writes:

Sometimes I regreted not fixing him a job. But really, who could tell whether it was a good decision or not? Maybe it was his destiny to endure and turn out to be great…. When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the north pole, a mountaineer who determined to scale Mt. Everest, and a traveler who would brave thunders and lightnings to reach his destination. Yitang never came back to me requesting recommendation letters. Apparently, he did not seek a job.

An interview with Zhang, conducted by Michael Segal and published in Nautilus, puts a somewhat different spin on the matter of recommendations. In the interview Zhang remarks: “During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor did not write me letters of recommendation.”

Zhang spent most of the next decade without an academic position. Here’s the Wikipedia account of that period: “Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for a New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop.” The article by Virginia Stuart makes clear that the job as an accountant and the job at Subway were the same job. “In a pinch, he would help out behind the counter, a fact that has been exaggerated in the press and has inspired online banter about a mathematical genius making sandwiches for a living.”

In 1999 Zhang was hired as a lecturer in the department of mathematics and statistics at UNH. I haven’t been able to learn exactly how this came about. The chairman of the department of the time was Kenneth Appel, who had his own history of breakthrough proofs—in his case, the four-color theorem. Appel died just a few days after learning of Zhang’s triumph.

UNH saved Zhang from a life of making footlong sandwiches, but for 15 years his job was a nontenured lectureship, on a year-to-year contract. He apparently taught a lot of first-year calculus—and got rave reviews on ratemyprofessors.com. One review says, in its entirety: “HOT!!!” The UNH web site still lists him as a lecturer, but news reports this month say he has been promoted to full professor.

At the JMM, Andrew Granville of Montreal gave a big tutorial talk summing up the recent and ongoing work on bounded gaps between primes. But the main event happened a day earlier, when both Maynard and Zhang spoke in the same session. The room was jammed. Maynard went first, and gave a lucid and precise account of his own work. When he finished, Zhang stepped forward with a sheaf of transparencies, which he’d been scribbling on with a Sharpie pen during some of the earlier talks in the session. There was a brief delay while the video projector was set aside and the overhead projector was moved into place and switched on. Still more people flowed into the room from the doors at the rear. (I had arrived early to get a seat up front.) The hubbub continued until Granville rose and bellowed out “Quiet!” Then Zhang turned to face the audience and announced: “I know you all expect me to talk about twin primes, as the abstract says, but I’ve decided to talk about something else.” I was mildly shocked. I don’t think I’ve ever seen anyone pull such a switcheroo at a major conference.

The JMM still provides an overhead projector in every lecture room. Among the talks I attended this year, Zhang was the only speaker who made use of this quaint device.

“Something else” turned out to be the Goldbach conjecture—the assertion that every even integer greater than 2 can be expressed as the sum of two prime numbers. Zhang wasn’t claiming to have proved the Goldbach conjecture; he wasn’t even laying out a clear pathway to a proof; but he did vaguely hint that he has some idea of how to proceed. We’ll see, I guess. For the moment he has a healthy balance of credibility.

An hour after he gave his talk, Zhang was awarded a share of the Frank Nelson Cole Prize in Number Theory.

Nonmathmatical geographic/autobiographic/photographic addendum. The JMM was held in Baltimore this year, a city I lived in from age 0 to age 4 and again from age 20 through 23. On this trip I spent my first few hours in town strolling around taking pictures, and belatedly discovered that my camera was set to one of the cute “illustration” modes that modern digital technology has to offer. I was shocked at this switcheroo, too, but after a few days the cartoonish pictures grew on me.

The image above shows the Morris A. Mechanic Theater, which was the centerpiece of an urban redevelopment project in the early 1970s, when I was a young man about town. The theater was supposed to bring Broadway tryouts and other forms of stagecraft back to the heart of the city. It closed in 2004, and now even the posters promising another phase of redevelopment are looking tattered.

As for the Hotel Junker/Envy Hotel, below, I didn’t stay there, but it was just around the corner. Echt Baltimore.

Posted in mathematics | 4 Comments