A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are 142857 × 1 = 142857 142857 × 2 = 285714 142857 × 3 = 428571 142857 × 4 = 571428 142857 × 5 = 714285 142857 × 6 = 857142 Details To qualify as a cyclic number, it is required that consecutive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, because even though all cyclic permutations are multiples, they are not consecutive integer multiples: 076923 × 1 = 076923 076923 × 3 = 230769 076923 × 4 = 307692 076923 × 9 = 692307 076923 × 10 = 769230 076923 × 12 = 923076 The following trivial cases are typically excluded: single digits, e.g.: 5 repeated digits, e.g.: 555 repeated cyclic numbers, e.g.: 142857142857 If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal, due to the necessary structure given in the next section. Allowing leading zeros, the sequence of cyclic numbers begins: (106 − 1) / 7 = 142857 (6 digits) (1016 − 1) / 17 = 0588235294117647 (16 digits) (1018 − 1) / 19 = 052631578947368421 (18 digits) (1022 − 1) / 23 = 0434782608695652173913 (22 digits) (1028 − 1) / 29 = 0344827586206896551724137931 (28 digits) (1046 − 1) / 47 = 0212765957446808510638297872340425531914893617 (46 digits) (1058 − 1) / 59 = 0169491525423728813559322033898305084745762711864406779661 (58 digits) (1060 − 1) / 61 = 016393442622950819672131147540983606557377049180327868852459 (60 digits) (1096 − 1) / 97 = 010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567 (96 digits) Relation to repeating decimals Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number. For example: 1/7 = 0.142857 142857... Multiples of these fractions exhibit cyclic permutation: 1/7 = 0.142857 142857... 2/7 = 0.285714 285714... 3/7 = 0.428571 428571... 4/7 = 0.571428 571428... 5/7 = 0.714285 714285... 6/7 = 0.857142 857142... Form of cyclic numbers From the relation to unit fractions, it can be shown that cyclic numbers are of the form of the Fermat quotient where b is the number base (10 for decimal), and p is a