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 carries into 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.”