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Carryless arithmetic base 10, mod 10 version

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Definition

Carryless arithmetic is a commutative algebra in which operations on single digits are done without carry:

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 recordkeeping.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.[1]

Units in carryless arithmetic mod 10

Define the units in carryless arithmetic mod 10 to be the numbers 1, 3, 7 and 9 (these divide any number).

Primes in carryless arithmetic mod 10

A prime is a number N, not a unit, whose only factorizations are of the form N = u * M, where u is a unit. There are two types: e-type primes (A163396) and f-type (A169984).

Sequences

The units in carryless arithmetic mod 10 are

 := {1, 3, 7, 9} = {1, 3, -3, -1}

The “evenish” numbers, in which all digits are even (Cf. A014263,) are

= {0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 200, 202, 204, 206, 208, 220, 222, 224, 226, 228, 240, 242, 244, 246, 248, ...}

The “fiveish” numbers, in which all digits are 0 or 5 (Cf. A169964,) are

= {0, 5, 50, 55, 500, 505, 550, 555, 5000, 5005, 5050, 5055, 5500, 5505, 5550, 5555, 50000, 50005, 50050, 50055, 50500, 50505, 50550, 50555, 55000, 55005, 55050, ...}

The zero-divisors (Cf. A169884) are

= {0, 2, 4, 5, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 50, 55, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 200, 202, 204, 206, 208, 220, 222, 224, 226, 228, 240, ...}

The positive numbers which are not zero divisors (Cf. A169968) are

= {1, 3, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 63, 65, ...}

The primes in carryless arithmetic mod 10 (Cf. A169887,) with index , are

= {21, 23, 25, 27, 29, 41, 43, 45, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 63, 65, 67, 69, 81, 83, 85, 87, 89, 201, 209, 227, 229, 241, 243, 261, 263, 287, 289, 403, 407, ...}

The squares in carryless arithmetic mod 10 (Cf. A059729) are

{0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 121, 144, 169, 186, 105, 126, 149, 164, 181, 400, 441, 484, 429, 466, 405, 446, 489, 424, 461, 900, 961, 924, 989, 946, 905, 966, 929, 984, ...}

See also

Notes

  1. [ALS]

References