## FANDOM

114 Pages

cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number.

For example, the 4-digit number 2497: (comes from 1/5 = 0.249724972497...)

2497 × 1 = 2497

2497 × 2 = 4972

2497 × 3 = 7249

2497 × 4 = 9724

Also the 6-digit number 186X35: (comes from 1/7 = 0.186X35186X35...)

186X35 × 1 = 186X35

186X35 × 2 = 35186X

186X35 × 3 = 5186X3

186X35 × 4 = 6X3518

186X35 × 5 = 86X351

186X35 × 6 = X35186

Note that {2497, 186X35} uses all digits except the smallest (0) and the largest (E) exactly once.

From the relation to unit fractions, it can be shown that cyclic numbers are of the form of the Fermat quotient (10^(p-1)-1)/p

Primes p that give cyclic numbers are called full reptend primes or long primes, these primes are

5, 7, 15, 27, 35, 37, 45, 57, 85, 87, 95, X7, E5, E7, 105, 107, 117, 125, 145, 167, 195, 1X5, 1E5, 1E7, 205, 225, 255, 267, 277, 285, 295, 315, 325, 365, 377, 397, 3X5, 3E5, 3E7, 415, 427, 435, 437, 447, 455, 465, 497, 4X5, 517, 527, 535, 545, 557, 565, 575, 585, 5E5, 615, 655, 675, 687, 695, 6X7, 705, 735, 737, 745, 767, 775, 785, 797, 817, 825, 835, 855, 865, 8E5, 8E7, 907, 927, 955, 965, 995, 9X7, 9E5, X07, X17, X35, X37, X45, X77, X87, X95, XE7, E25, E37, E45, E95, E97, EX5, EE5, EE7, ...

All of these primes end with either 5 or 7, the known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that 10 is a primitive root modulo p. Artin's conjecture on primitive roots is that this sequence contains 45.X242..% of the primes. (Artin's constant CArtin is 0.45X242...)

By Midy's theorem, we have 24+97=EE, 186+X35=EEE, 18+6X+35=EE, ...

Community content is available under CC-BY-SA unless otherwise noted.