Cyclic number

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 &times; 1 = 142857
 * 142857 &times; 2 = 285714
 * 142857 &times; 3 = 428571
 * 142857 &times; 4 = 571428
 * 142857 &times; 5 = 714285
 * 142857 &times; 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 &times; 1 = 076923
 * 076923 &times; 3 = 230769
 * 076923 &times; 4 = 307692
 * 076923 &times; 9 = 692307
 * 076923 &times; 10 = 769230
 * 076923 &times; 12 = 923076

The following trivial cases are typically excluded:
 * 1) single digits, e.g.: 5
 * 2) repeated digits, e.g.: 555
 * 3) 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 −&thinsp;1) / 7 = 142857 (6 digits)
 * (1016 −&thinsp;1) / 17 = 0588235294117647 (16 digits)
 * (1018 −&thinsp;1) / 19 = 052631578947368421 (18 digits)
 * (1022 −&thinsp;1) / 23 = 0434782608695652173913 (22 digits)
 * (1028 −&thinsp;1) / 29 = 0344827586206896551724137931 (28 digits)
 * (1046 −&thinsp;1) / 47 = 0212765957446808510638297872340425531914893617 (46 digits)
 * (1058 −&thinsp;1) / 59 = 0169491525423728813559322033898305084745762711864406779661 (58 digits)
 * (1060 −&thinsp;1) / 61 = 016393442622950819672131147540983606557377049180327868852459 (60 digits)
 * (1096 −&thinsp;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


 * $$\frac{b^{p-1}-1}{p}$$

where b is the number base (10 for decimal), and p is a prime that does not divide b. (Primes p that give cyclic numbers in base b are called full reptend primes or long primes in base b).

For example, the case b = 10, p = 7 gives the cyclic number 142857, and the case b = 12, p = 5 gives the cyclic number 2497.

Not all values of p will yield a cyclic number using this formula; for example, the case b = 10, p = 13 gives 076923076923, and the case b = 12, p = 19 gives 076B45076B45076B45. These failed cases will always contain a repetition of digits (possibly several).

The first values of p for which this formula produces cyclic numbers in decimal (b = 10) are


 * 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983, ...

For b = 12 (duodecimal), these ps are
 * 5, 7, 17, 31, 41, 43, 53, 67, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, 223, 257, 269, 281, 283, 293, 317, 353, 367, 379, 389, 401, 449, 461, 509, 523, 547, 557, 569, 571, 593, 607, 617, 619, 631, 641, 653, 691, 701, 739, 751, 761, 773, 787, 797, 809, 821, 857, 881, 929, 953, 967, 977, 991, ...

For b = 2 (binary), these ps are
 * 3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797, 821, 827, 829, 853, 859, 877, 883, 907, 941, 947, ...

For b = 3 (ternary), these ps are
 * 2, 5, 7, 17, 19, 29, 31, 43, 53, 79, 89, 101, 113, 127, 137, 139, 149, 163, 173, 197, 199, 211, 223, 233, 257, 269, 281, 283, 293, 317, 331, 353, 379, 389, 401, 449, 461, 463, 487, 509, 521, 557, 569, 571, 593, 607, 617, 631, 641, 653, 677, 691, 701, 739, 751, 773, 797, 809, 811, 821, 823, 857, 859, 881, 907, 929, 941, 953, 977, ...

There are no such ps in the hexadecimal system.

The known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that b is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains 37.395..% of the primes (for b in ).

Construction of cyclic numbers
Cyclic numbers can be constructed by the following procedure:

Let b be the number base (10 for decimal) Let p be a prime that does not divide b. Let t = 0. Let r = 1. Let n = 0. loop:
 * Let t = t +&thinsp;1
 * Let x = r &sdot; b
 * Let d = int(x / p)
 * Let r = x mod p
 * Let n = n &sdot; b + d
 * If r ≠&thinsp;1 then repeat the loop.

if t = p − 1 then n is a cyclic number.

This procedure works by computing the digits of 1/p in base b, by long division. r is the remainder at each step, and d is the digit produced.

The step


 * n = n &sdot; b + d

serves simply to collect the digits. For computers not capable of expressing very large integers, the digits may be output or collected in another way.

If t ever exceeds p/2, then the number must be cyclic, without the need to compute the remaining digits.

Properties of cyclic numbers

 * When multiplied by their generating prime, the result is a sequence of b − 1 digits, where b is the base (e.g. 10 in decimal). For example, in decimal, 142857 × 7 = 999999.
 * When split into groups of equal length (of two, three, four, etc... digits), and the groups are added, the result is a sequence of b - 1 digits. For example, 14 + 28 + 57 = 99, 142 + 857 = 999, 1428 + 5714+ 2857 = 9999, etc. ... This is a special case of Midy's Theorem.
 * All cyclic numbers are divisible by b − 1 where b is the base (e.g. 9 in decimal) and the sum of the remainder is a multiple of the divisor. (This follows from the previous point.)

Other numeric bases
Using the above technique, cyclic numbers can be found in other numeric bases. (Not all of these follow the second rule (all successive multiples being cyclic permutations) listed in the Special Cases section above) In each of these cases, the digits across half the period add up to the base minus one. Thus for binary, the sum of the bits across half the period is 1; for ternary, it is 2, and so on.

In binary, the sequence of cyclic numbers begins:
 * 11 (3) → 01
 * 101 (5) → 0011
 * 1011 (11) → 0001011101
 * 1101 (13) → 000100111011
 * 10011 (19) → 000011010111100101
 * 11101 (29) → 0000100011010011110111001011

In ternary:
 * 2 (2) → 1
 * 12 (5) → 0121
 * 21 (7) → 010212
 * 122 (17) → 0011202122110201
 * 201 (19) → 001102100221120122

In quaternary, there are none.

In quinary:
 * 2 (2) → 2
 * 3 (3) → 13
 * 12 (7) → 032412
 * 32 (17) → 0121340243231042
 * 43 (23) → 0102041332143424031123
 * 122 (37) → 003142122040113342441302322404331102

In senary:
 * 15 (11) → 0313452421
 * 21 (13) → 024340531215
 * 25 (17) → 0204122453514331
 * 105 (41) → 0051335412440330234455042201431152253211
 * 135 (59) → 0033544402235104134324250301455220111533204514212313052541
 * 141 (61) → 003312504044154453014342320220552243051511401102541213235335

In base 7:
 * 2 (2) → 3
 * 5 (5) → 1254
 * 14 (11) → 0431162355
 * 16 (13) → 035245631421
 * 23 (17) → 0261143464055232
 * 32 (23) → 0206251134364604155323

In octal:
 * 3 (3) → 25
 * 5 (5) → 1463
 * 13 (11) → 0564272135
 * 35 (29) → 0215173454106475626043236713
 * 65 (53) → 0115220717545336140465103476625570602324416373126743
 * 73 (59) → 0105330745756511606404255436276724470320212661713735223415

In nonary, the unique cyclic number is
 * 2 (2) → 4

In base 11:
 * 2 (2) → 5
 * 3 (3) → 37
 * 12 (13) → 093425A17685
 * 16 (17) → 07132651A3978459
 * 21 (23) → 05296243390A581486771A
 * 27 (29) → 04199534608387A69115764A2723

In duodecimal:
 * 5 (5) → 2497
 * 7 (7) → 186A35
 * 15 (17) → 08579214B36429A7
 * 27 (31) → 0478AA093598166B74311B28623A55
 * 35 (41) → 036190A653277397A9B4B85A2B15689448241207
 * 37 (43) → 0342295A3AA730A068456B879926181148B1B53765

In ternary (b = 3), the case p = 2 yields 1 as a cyclic number. While single digits may be considered trivial cases, it may be useful for completeness of the theory to consider them only when they are generated in this way.

It can be shown that no cyclic numbers (other than trivial single digits, i.e. p = 2) exist in any numeric base which is a perfect square, that is, base 4, 9, 16, 25, etc.