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| {{about|numbers where permutations of their digits (in some base) yield related numbers|the number theoretic concept|cyclic number (group theory)}}
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| A '''cyclic number''' is an [[integer]] in which [[cyclic permutation]]s of the digits are successive multiples of the number. The most widely known is [[142857 (number)|142857]]:
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| :142857 × 1 = 142857
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| :142857 × 2 = 285714
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| :142857 × 3 = 428571
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| :142857 × 4 = 571428
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| :142857 × 5 = 714285
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| :142857 × 6 = 857142
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| == Details ==
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| To qualify as a cyclic number, it is required that successive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, even though all cyclic permutations are multiples:
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| :076923 × 1 = 076923
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| :076923 × 3 = 230769
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| :076923 × 4 = 307692
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| :076923 × 9 = 692307
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| :076923 × 10 = 769230
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| :076923 × 12 = 923076
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| The following trivial cases are typically excluded:
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| #single digits, e.g.: 5
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| #repeated digits, e.g.: 555
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| #repeated cyclic numbers, e.g.: 142857142857
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| 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:
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| :(10<sup>6</sup>-1) / '''7''' = 142857 (6 digits)
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| :(10<sup>16</sup>-1) / '''17''' = 0588235294117647 (16 digits)
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| :(10<sup>18</sup>-1) / '''19''' = 052631578947368421 (18 digits)
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| :(10<sup>22</sup>-1) / '''23''' = 0434782608695652173913 (22 digits)
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| :(10<sup>28</sup>-1) / '''29''' = 0344827586206896551724137931 (28 digits)
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| :(10<sup>46</sup>-1) / '''47''' = 0212765957446808510638297872340425531914893617 (46 digits)
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| :(10<sup>58</sup>-1) / '''59''' = 0169491525423728813559322033898305084745762711864406779661 (58 digits)
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| :(10<sup>60</sup>-1) / '''61''' = 016393442622950819672131147540983606557377049180327868852459 (60 digits)
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| == Relation to repeating decimals ==
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| Cyclic numbers are related to the [[Repeating decimal|recurring digital representations]] of [[unit fractions]]. A cyclic number of length ''L'' is the digital representation of
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| :1/(''L'' + 1).
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| Conversely, if the digital period of 1 /''p'' (where ''p'' is prime) is
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| :''p'' − 1,
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| then the digits represent a cyclic number.
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| For example:
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| :1/7 = 0.142857 142857….
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| Multiples of these fractions exhibit cyclic permutation:
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| :1/7 = 0.142857 142857…
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| :2/7 = 0.285714 285714…
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| :3/7 = 0.428571 428571…
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| :4/7 = 0.571428 571428…
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| :5/7 = 0.714285 714285…
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| :6/7 = 0.857142 857142….
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| == Form of cyclic numbers ==
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| From the relation to unit fractions, it can be shown that cyclic numbers are of the form
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| :<math>\frac{b^{p-1}-1}{p}</math>
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| where ''b'' is the [[Radix|number base]] (10 for [[decimal]]), and ''p'' is a [[Prime number|prime]] that does not [[Divisor|divide]] ''b''. (Primes ''p'' that give cyclic numbers are called [[full reptend prime]]s or long primes).
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| For example, the case ''b'' = 10, ''p'' = 7 gives the cyclic number 142857.
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| Not all values of ''p'' will yield a cyclic number using this formula; for example ''p''=13 gives 076923076923. These failed cases will always contain a repetition of digits (possibly several).
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| The first values of ''p'' for which this formula produces cyclic numbers in decimal are (sequence [[OEIS:A001913|A001913]] in [[On-Line Encyclopedia of Integer Sequences|OEIS]]):
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| :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 …
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| 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 n|primitive root modulo p]]. A [[Artin's conjecture on primitive roots|conjecture of Emil Artin]] <ref>http://mathworld.wolfram.com/ArtinsConstant.html</ref> is that this sequence contains 37.395..% of the primes.
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| == Construction of cyclic numbers ==
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| Cyclic numbers can be constructed by the following [[Algorithm|procedure]]:
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| Let ''b'' be the number base (10 for decimal)<br>
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| Let ''p'' be a prime that does not divide ''b''.<br>
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| Let ''t'' = 0.<br>
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| Let ''r'' = 1.<br>
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| Let ''n'' = 0.<br>
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| loop:
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| :Let ''t'' = ''t'' + 1
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| :Let ''x'' = ''r'' · ''b''
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| :Let ''d'' = [[Floor function|int]](''x'' / ''p'')
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| :Let ''r'' = ''x'' [[modulo operation|mod]] ''p''
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| :Let ''n'' = ''n'' · ''b'' + ''d''
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| :If ''r'' ≠ 1 then repeat the loop.
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| if ''t'' = ''p'' − 1 then ''n'' is a cyclic number.
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| 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.
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| The step
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| :''n'' = ''n'' · ''b'' + ''d''
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| 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.
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| Note that if ''t'' ever exceeds ''p''/2, then the number must be cyclic, without the need to compute the remaining digits.
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| == Properties of cyclic numbers == | |
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| *When multiplied by their generating prime, results in a sequence of {{'}}''base''−1' digits (9 in decimal). ''Decimal 142857 × 7 = 999999.''
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| *When split in two,three four etc...regarding base 10,100,1000 etc.. by its digits and added the result is a sequence of 9's. ''14 + 28 + 57 = 99'', ''142 + 857 = 999'', ''1428 + 5714+ 2857 = 9999'' etc ... (This is a special case of [[Midy's Theorem]].)
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| *All cyclic numbers are divisible by {{'}}''base''−1' (9 in decimal) and the sum of the remainder is the a multiple of the divisor. (This follows from the previous point.)
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| == Other numeric bases ==
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| Using the above technique, cyclic numbers can be found in other numeric bases. (Note that not all of these follow the second rule (all successive multiples being cyclic permutations) listed in the Special Cases section above)
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| In [[Binary numeral system|binary]], the sequence of cyclic numbers begins:
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| :11 (3) → 01
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| :101 (5) → 0011
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| :1011 (11) → 0001011101
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| :1101 (13) → 000100111011
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| :10011 (19) → 000011010111100101
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| In [[Ternary numeral system|ternary]]:
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| :12 (5) → 0121
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| :21 (7) → 010212
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| :122 (17) → 0011202122110201
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| :201 (19) → 001102100221120122
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| :1002 (29) → 0002210102011122200121202111
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| In [[Quaternary numeral system|quaternary]]:
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| : none
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| In [[quinary]]:
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| :3 (3) → 13
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| :12 (7) → 032412
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| :32 (17) → 0121340243231042
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| :122 (37) → 003142122040113342441302322404331102
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| :133 (43) → 002423141223434043111442021303221010401333
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| In [[senary]]:
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| :15 (11) → 0313452421
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| :21 (13) → 024340531215
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| :25 (17) → 0204122453514331
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| :31 (19) → 015211325015211325
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| :105 (41) → 0051335412440330234455042201431152253211
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| In [[septenary]]:
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| :5 (5) → 1254
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| :14 (11) → 0431162355
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| :16 (13) → 035245631421
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| :23 (17) → 0261143464055232
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| :32 (23) → 0206251134364604155323
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| In [[octal]]:
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| :3 (3) → 25
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| :5 (5) → 1463
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| :13 (11) → 0564272135
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| :35 (29) → 0215173454106475626043236713
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| :65 (53) → 0115220717545336140465103476625570602324416373126743
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| In [[nonary]]:
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| : none
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| In [[Base 11]]:
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| :3 (3) → 37
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| :12 (13) → 093425A17685
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| :16 (17) → 07132651A3978459
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| :21 (23) → 05296243390A581486771A
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| :27 (29) → 04199534608387A69115764A2723
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| In [[duodecimal]]:
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| :5 (5) → 2497
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| :7 (7) → 186A35
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| :15 (17) → 08579214B36429A7
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| :27 (31) → 0478AA093598166B74311B28623A55
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| :35 (41) → 036190A653277397A9B4B85A2B15689448241207
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| In [[Base 13]]:
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| :5 (5) → 27A5
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| :B (11) → 12495BA837
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| :16 (19) → 08B82976AC414A3562
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| :25 (31) → 055B42692C21347C7718A63A0AB985
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| In [[Tetradecimal|Base 14]]:
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| :3 (3) → 49
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| :13 (17) → 0B75A9C4D2683419
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| :15 (19) → 0A45C7522D398168BB
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| In [[Base 15]]:
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| :D (13) → 124936DCA5B8
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| :14 (19) → 0BC9718A3E3257D64B
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| :18 (23) → 09BB1487291E533DA67C5D
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| In [[hexadecimal]]:
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| : none
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| In Base 17:
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| :3 (3) → 5B
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| :5 (5) → 36DA
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| :7 (7) → 274E9C
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| :B (11) → 194ADF7C63
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| In Base 18:
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| :B (11) → 1B834H69ED
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| :1B (29) → 0B31F95A9GDAE4H6EG28C781463D
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| :21 (37) → 08DB37565F184FA3G0H946EACBC2G9D27E1H
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|
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| In Base 19:
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| :7 (7) → 2DAG58
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| :B (11) → 1DFA6H538C
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| :D (13) → 18EBD2HA475G
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| In [[vigesimal|Base 20]]:
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| :3 (3) → 6D
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| :D (13) → 1AF7DGI94C63
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| :H (17) → 13ABF5HCIG984E27
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| In Base 21:
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| :J (19) → 1248HE7F9JIGC36D5B
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| :12 (23) → 0J3DECG92FAK1H7684BI5A
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| :18 (29) → 0F475198EA2IH7K5GDFJBC6AI23D
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| In Base 22:
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| :5 (5) → 48HD
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| :H (17) → 16A7GI2CKFBE53J9
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| :J (17) → 13A95H826KIBCG4DJF
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| In Base 23:
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| :3 (3) → 7F
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| :5 (5) → 4DI9
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| :H (17) → 182G59AILEK6HDC4
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| In [[Base 24]]:
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| :7 (7) → 3A6KDH
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| :B (11) → 248HALJF6D
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| :D (13) → 1L795CM3GEIB
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| :H (17) → 19L45FCGME2JI8B7
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| Note that 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.
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| It can be shown that no cyclic numbers (other than trivial single digits) exist in any numeric base which is a [[Square number|perfect square]]; thus there are no cyclic numbers in [[hexadecimal]], [[base 4]], or [[nonary]].
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| == See also ==
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| *[[Repeating decimal]]
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| *[[Fermat's little theorem]]
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| *[[Cyclic permutation of integer]]
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| *[[Parasitic number]]
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| == References ==
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| {{Reflist}}
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| ==Further reading==
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| *Gardner, Martin. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments From Scientific American. New York: The Mathematical Association of America, 1979. pp. 111-122.
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| *Kalman, Dan; 'Fractions with Cycling Digit Patterns' The College Mathematics Journal, Vol. 27, No. 2. (Mar., 1996), pp. 109-115.
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| * Leslie, John. ''"The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of ...."'', Longman, Hurst, Rees, Orme, and Brown, 1820, ISBN 1-4020-1546-1
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| *Wells, David; ''"[[The Penguin Dictionary of Curious and Interesting Numbers]]"'', Penguin Press. ISBN 0-14-008029-5
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| ==External links==
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| * {{MathWorld | urlname=CyclicNumber | title=Cyclic Number}}
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| * [http://www.youtube.com/watch?v=WUlaUalgxqI Youtube: "Cyclic Numbers - Numberphile"]
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| [[Category:Number theory]]
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| [[Category:Permutations]]
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