Carryless arithmetic base 10, mod 10 version

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.

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

$\scriptstyle {\mathcal {U}}\,$ := {1, 3, 7, 9} = {1, 3, -3, -1}

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

$\scriptstyle {\mathcal {E}}\,$ = {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

$\scriptstyle {\mathcal {F}}\,$ = {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

$\scriptstyle {\mathcal {Z}}\,:=\,{\mathcal {E}}\,\cup \,{\mathcal {F}}\,$ = {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

$\scriptstyle {\mathcal {N}}\,$ = {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 $\scriptstyle n\,\geq \,1\,$, are

$\scriptstyle {\mathcal {P}}\,$ = {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 $\scriptstyle n\,\otimes \,n,\ n\,\geq \,0,\,$ 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

• Index entries for sequences related to carryless arithmetic
• A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

• Dismal arithmetic

References

• [ALS] David Applegate, Marc LeBrun, N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version, 2010.