Easy as abc

It all starts with the equation a + b = c, which looks straightforward enough. Assume that a, b and c are positive integers that have no divisors in common other than 1; for example, the triple {1, 2, 3} meets this condition, and so does {4, 5, 9}.

Now let’s take the product of the three numbers abc, list all the prime factors of this product, and cast out any duplicates, so that each prime in the list appears just once. The product of these unique primes is called the radical of abc; it is necessarily a “square-free” number, one that cannot be divided by any perfect square. For the triple {1, 2, 3} the product is 1 × 2 × 3 = 6 and the prime factor list is (2, 3); since this list has no duplicates, the radical rad(6) is 6. In the case of {4, 5, 9}, the product is 4 × 5 × 9 = 180 and the factor list is (2, 2, 3, 3, 5). Removing the duplicated 2s and 3s leaves the unique factor list (2, 3, 5), so that rad(180) = 30.

Note that in both of these examples, rad(abc) > c. Can it ever happen otherwise? Can rad(abc) be less than c? Yes, it can. The triple {5, 27, 32} has the product 5 × 27 × 32 = 4320; it’s easy to see that the unique primes in this product are again (2, 3, 5), and so rad(4320) = 30, which is less than c = 32. Triples with this property are called abc-hits. There are infinitely many of them, and yet they are quite rare. Among all abc triples with c ≤ 10000, there are just 120 abc-hits.

If rad(abc) can be less than c, just how much less? Can we find an example where rad(abc) is so much smaller that the square of rad(abc) is also less than c? No one knows of such a triple, but there are some examples where [rad(abc)]^α < c for some exponent α greater than 1 though less than 2. Take the triple {2, 6436341, 6436343}. Here b is equal to 3¹⁰ × 109 and c is equal to 23⁵, and so rad(abc) = 2 × 3 × 23 × 109 = 15042. And 15042^α < 6436343 for any α < 1.6299.

So who cares? Number theorists, that’s who.

In 1985 Joseph Oesterlé of the University of Paris and David W. Masser of the University of Basel conjectured that there are only finitely many such exceptional triples. Given a positive number ε that can be made arbitrarily small, [rad(abc)]^{1 + ε} can be less than c only in a finite number of cases (which will depend on ε). That’s the abc conjecture. Although it may seem like a rather arbitrary game with numbers, if it turns out to be true, there’s a long list of important consequences.

Why am I going on about this just now? A friend has passed along a rumor that we may soon hear big news about the abc conjecture.

Some resources:

• The ABC Conjecture Home Page, by Abderrahmane Nitaj
• The Amazing ABC Conjecture, by Ivars Peterson
• ABC@home, a project led by Hendrik W. Lenstra Jr., B. de Smit and W. J. Palenstijn
• The ABC Conjecture, a presentation by Frits Beukers (PDF).
• It’s As Easy as ABC, a Notices article by Andrew Granville and Thomas J. Tucker.

Update 2007-05-26: Time to lift the cloak of mystery. At a conference held last week at Columbia University, Lucien Szpiro of the City University of New York gave the last talk on the last day, and quietly let it be known that he had proved at least part of the abc conjecture. I’m grateful to Kevin O’Bryant for giving me an early heads-up on this news. There’s now a little more on Peter Woit’s blog.