The fabled carefree residents of the Carryless Islands in the remote South Pacific have very few possessions, which is just as well, since their notion of arithmetic is ill-suited to accurate record-keeping. When they add or multiply numbers, they follow similar rules to ours, except that there are

no carriesinto other digit positions. Addition and multiplication of single-digit numbers are performed by a process that we would call “reduction mod 10.” Any carry digits are simply ignored. So 9+4 = 3, 5+5 = 0, 9×4 = 6, 5×4 = 0, and so on.

With this fable, David Applegate, Marc LeBrun and N. J. A. Sloane introduce a new scheme of arithmetic in a paper newly posted on the arXiv.

And if you think the mathematics sounds trivial, try explaining the structure of this sequence:

21, 23, 25, 27, 29, 41, 43, 45, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 63,

which lists the first 20 “carryless primes.”

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