A cyclic number is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

for which

cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k ...

s 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 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

, 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

) 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 In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as :q_p(a) = \frac, or :\delta_p(a) = \frac. This article is about the former; for the latter see ''p''-derivation. The quotient is named a ...

\frac

where ''b'' is the number base (10 for

), and ''p'' is a

that does not divide ''b''. (Primes ''p'' that give cyclic numbers in base ''b'' are called

full reptend prime In number theory, a full reptend prime, full repetend prime, proper primeDickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or long prime in base ''b'' is an odd prime number ''p'' such that the Fermat ...

s 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

(''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 The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is i ...

), these ''p''s 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 Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical op ...

), these ''p''s 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 ''p''s 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 ''p''s in the

hexadecimal Hexadecimal (also known as base-16 or simply hex) is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbo ...

system. The known pattern to this sequence comes from

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

, 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'' + 1 :Let ''x'' = ''r'' ⋅ ''b'' :Let ''d'' = int(''x'' / ''p'') :Let ''r'' = ''x'' mod ''p'' :Let ''n'' = ''n'' ⋅ ''b'' + ''d'' :If ''r'' ≠ 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 In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals (positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier step ...

. ''r'' is the

remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In a ...

at each step, and ''d'' is the digit produced. The step :''n'' = ''n'' ⋅ ''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

, 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 The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS), as well as the current and most recent of the twelve periods of the ...

, 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 A senary () numeral system (also known as base-6, heximal, or seximal) has 6, six as its radix, base. It has been adopted independently by a small number of cultures. Like the decimal base 10, the base is a semiprime, though it is unique as the p ...

: :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 Octal (base 8) is a numeral system with eight as the base. In the decimal system, each place is a power of ten. For example: : \mathbf_ = \mathbf \times 10^1 + \mathbf \times 10^0 In the octal system, each place is a power of eight. For ex ...

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

nonary A ternary numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log2 3 (about 1.58496) bits of information. Although ''ternary'' ...

, 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

: :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.

References

External links

* * {{Classes of natural numbers Number theory Permutations