A cyclic number is an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

for which

cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...

s of the digits are successive integer multiples of the number. The most widely known is the six-digit number

142857 The number 142,857 is a Kaprekar number. 142857, the six repeating digits of (0.), is the best-known cyclic number in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond t ...

, 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 of the Hindu–Arabic numeral ...

, 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 :

\frac

where ''b'' is the

number base In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...

(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 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...

), 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), 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 In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, he ...

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 o ...

, 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 steps ...

. ''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 algeb ...

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. 9 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 9s. For example, 14 + 28 + 57 = 99, 142 + 857 = 999, 1428 + 5714+ 2857 = 9999, etc. ... This is a special case of

Midy's Theorem In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions ''a''/''p'' where ''p'' is a prime and ''a''/''p'' has a repeating decimal expansion with an even period . If the p ...

. *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 :100101 (37) → 00000110101011100101111100101010001101 :110101 (53) → 00000100101101001111001001101101111101101001011000011011001001 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). It follows the Neogene Period and spans from 2.58 million year ...

: :(none) In

quinary Quinary (base-5 or pental) is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand. In the quinary place system, five numerals, from 0 to 4, are used to represent a ...

: :2 (2) → 2 :3 (3) → 13 :12 (7) → 032412 :32 (17) → 0121340243231042 :43 (23) → 0102041332143424031123 :122 (37) → 003142122040113342441302322404331102 :133 (43) → 002423141223434043111442021303221010401333 In

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

: :15 (11) → 0313452421 :21 (13) → 024340531215 :25 (17) → 0204122453514331 :105 (41) → 0051335412440330234455042201431152253211 :135 (59) → 0033544402235104134324250301455220111533204514212313052541 :141 (61) → 003312504044154453014342320220552243051511401102541213235335 :211 (79) → 002422325434441304033512354102140052450553133230121114251522043201453415503105 In base 7: :2 (2) → 3 :5 (5) → 1254 :14 (11) → 0431162355 :16 (13) → 035245631421 :23 (17) → 0261143464055232 :32 (23) → 0206251134364604155323 :56 (41) → 0112363262135202250565543034045314644161 In

octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...

: :3 (3) → 25 :5 (5) → 1463 :13 (11) → 0564272135 :35 (29) → 0215173454106475626043236713 :65 (53) → 0115220717545336140465103476625570602324416373126743 :73 (59) → 0105330745756511606404255436276724470320212661713735223415 :123 (83) → 0061262710366576352321570224030531344173277165150674112014254562075537472464336045 In

nonary A ternary numeral system (also called base 3 or trinary) has 3 (number), three as its radix, base. Analogous to a bit, a ternary numerical digit, digit is a trit (trinary digit). One trit is equivalent to binary logarithm, log2 3 (about 1.5 ...

: :2 (2) → 4 :(no others) In base 11: :2 (2) → 5 :3 (3) → 37 :12 (13) → 093425A17685 :16 (17) → 07132651A3978459 :21 (23) → 05296243390A581486771A :27 (29) → 04199534608387A69115764A2723 :29 (31) → 039A32146818574A71078964292536 In

: :5 (5) → 2497 :7 (7) → 186A35 :15 (17) → 08579214B36429A7 :27 (31) → 0478AA093598166B74311B28623A55 :35 (41) → 036190A653277397A9B4B85A2B15689448241207 :37 (43) → 0342295A3AA730A068456B879926181148B1B53765 :45 (53) → 02872B3A23205525A784640AA4B9349081989B6696143757B117 In base 13: :2 (2) → 6 :5 (5) → 27A5 :B (11) → 12495BA837 :16 (19) → 08B82976AC414A3562 :25 (31) → 055B42692C21347C7718A63A0AB985 :2B (37) → 0474BC3B3215368A25C85810919AB79642A7 :32 (41) → 04177C08322B13645926C8B550C49AA1B96873A6 In base 14: :3 (3) → 49 :13 (17) → 0B75A9C4D2683419 :15 (19) → 0A45C7522D398168BB :19 (23) → 0874391B7CAD569A4C2613 :21 (29) → 06A89925B163C0D73544B82C7A1D :3B (53) → 039AB8A075793610B146C21828DA43253D6864A7CD2C971BC5B5 :43 (59) → 03471937B8ACB5659A2BC15D09D74DA96C4A62531287843B21C80D4069 In base 15: :2 (2) → 7 :D (13) → 124936DCA5B8 :14 (19) → 0BC9718A3E3257D64B :18 (23) → 09BB1487291E533DA67C5D :1E (29) → 07B5A528BD6ACDE73949C6318421 :27 (37) → 061339AE2C87A8194CE8DBB540C26746D5A2 :2B (41) → 0574B51C68BA922DD80AE97A39D286345CC116E4 In

: :(none) In base 17: :2 (2) → 8 :3 (3) → 5B :5 (5) → 36DA :7 (7) → 274E9C :B (11) → 194ADF7C63 :16 (23) → 0C9A5F8ED52G476B1823BE :1E (31) → 09583E469EDC11AG7B8D2CA7234FF6 In base 18: :5 (5) → 3AE7 :B (11) → 1B834H69ED :1B (29) → 0B31F95A9GDAE4H6EG28C781463D :21 (37) → 08DB37565F184FA3G0H946EACBC2G9D27E1H :27 (43) → 079B57H2GD721C293DEBCHA86CA0F14AFG5F8E4365 :2H (53) → 0620C41682CG57EAFB3D4788EGHBFH5DGB9F51CA3726E4DA9931 :35 (59) → 058F4A6CEBAC3BG30G89DD227GE0AHC92D7B53675E61EH19844FFA13H7 In base 19: :2 (2) → 9 :7 (7) → 2DAG58 :B (11) → 1DFA6H538C :D (13) → 18EBD2HA475G :14 (23) → 0FD4291C784I35EG9H6BAE :1A (29) → 0C89FDE7G73HD1I6A9354B2BF15H :1I (37) → 09E73B5C631A52AEGHI94BF7D6CFH8DG8421 In base 20: :3 (3) → 6D :D (13) → 1AF7DGI94C63 :H (17) → 13ABF5HCIG984E27 :13 (23) → 0H7GA8DI546J2C39B61EFD :1H (37) → 0AG469EBHGF2E11C8CJ93FDA58234H5II7B7 :23 (43) → 0960IC1H43E878GEHD9F6JADJ17I2FG5BCB3526A4D :27 (47) → 08A4522B15ACF67D3GBI5J2JB9FEHH8IE974DC6G381E0H In base 21: :2 (2) → A :J (19) → 1248HE7F9JIGC36D5B :12 (23) → 0J3DECG92FAK1H7684BI5A :18 (29) → 0F475198EA2IH7K5GDFJBC6AI23D :1A (31) → 0E4FC4179A382EIK6G58GJDBAHCI62 :2B (53) → 086F9AEDI4FHH927J8F13K47B1KCE5BA672G533BID1C5JH0GD9J :38 (71) → 06493BB50C8I721A13HFE42K27EA785J4F7KEGBH99FK8C2DIJAJH356GI0ID6ADCF1G5D In base 22: :5 (5) → 48HD :H (17) → 16A7GI2CKFBE53J9 :J (19) → 13A95H826KIBCG4DJF :19 (31) → 0FDAE45EJJ3C194L68B7HG722I9KCH :1F (37) → 0D1H57G143CAFA2872L8K4GE5KHI9B6BJDEJ :1J (41) → 0BHFC7B5JIH3GDKK8CJ6LA469EAG234I5811D92F :23 (47) → 0A6C3G897L18JEB5361J44ELBF9I5DCE0KD27AGIFK2HH7 In base 23: :2 (2) → B :3 (3) → 7F :5 (5) → 4DI9 :H (17) → 182G59AILEK6HDC4 :21 (47) → 0B5K1AHE496JD4KCGEFF3L0MBH2LC58IDG39I2A6877J1M :2D (59) → 08M51CJK65AC1LJ27I79846E9H3BFME0HLA32GHCAL13KF4FDEIG8D5JB7 :3K (89) → 05LG6ADG0BK9CL4910HJ2J8I21CF5FHD4327B8C3864EMH16GC96MB2DA1IDLM53K3E4KLA7H759IJKFBEAJEGI8 In base 24: :7 (7) → 3A6KDH :B (11) → 248HALJF6D :D (13) → 1L795CM3GEIB :H (17) → 19L45FCGME2JI8B7 :17 (31) → 0IDMAK327HJ8C96N5A1D3KLG64FBEH :1D (37) → 0FDEM1735K2E6BG54CN8A91MGKI3L9HC7IJB :1H (41) → 0E14284G98IHDB2M5KBGN9MJLFJ7EF56ACL1I3C7 In base 25: :2 (2) → C :(no others) 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

*
Youtube: "Cyclic Numbers - Numberphile"
{{Classes of natural numbers Number theory Permutations