Cyclic number
A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. The most widely known is 142857:
- 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 successive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, even though all cyclic permutations are 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)
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
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 are called full reptend primes or long primes).
For example, the case b = 10, p = 7 gives the cyclic number 142857.
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).
The first values of p for which this formula produces cyclic numbers in decimal are (sequence A001913 in OEIS):
- 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 …
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 p. A conjecture of Emil Artin [1] is that this sequence contains 37.395..% of the primes.
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. r is the remainder 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.
Note that 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, results in a sequence of 'base-1' digits (9 in decimal). Decimal 142857 × 7 = 999999.
- 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.)
- 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.)
Other numeric bases
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)
In binary, the sequence of cyclic numbers begins:
- 11 (3) → 01
- 101 (5) → 0011
- 1011 (11) → 0001011101
- 1101 (13) → 000100111011
- 10011 (19) → 000011010111100101
In ternary:
- 12 (5) → 0121
- 21 (7) → 010212
- 122 (17) → 0011202122110201
- 201 (19) → 001102100221120122
- 1002 (29) → 0002210102011122200121202111
In quaternary:
- none
In quinary:
- 3 (3) → 13
- 12 (7) → 032412
- 32 (17) → 0121340243231042
- 122 (37) → 003142122040113342441302322404331102
- 133 (43) → 002423141223434043111442021303221010401333
In senary:
- 15 (11) → 0313452421
- 21 (13) → 024340531215
- 25 (17) → 0204122453514331
- 31 (19) → 015211325015211325
- 105 (41) → 0051335412440330234455042201431152253211
In septenary:
- 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
In nonary:
- none
In Base 11:
- 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
In Base 13:
- 5 (5) → 27A5
- B (11) → 12495BA837
- 16 (19) → 08B82976AC414A3562
- 25 (31) → 055B42692C21347C7718A63A0AB985
In Base 14:
- 3 (3) → 49
- 13 (17) → 0B75A9C4D2683419
- 15 (19) → 0A45C7522D398168BB
In Base 15:
- D (13) → 124936DCA5B8
- 14 (19) → 0BC9718A3E3257D64B
- 18 (23) → 09BB1487291E533DA67C5D
In hexadecimal:
- none
In Base 17:
- 3 (3) → 5B
- 5 (5) → 36DA
- 7 (7) → 274E9C
- B (11) → 194ADF7C63
In Base 18:
- B (11) → 1B834H69ED
- 1B (29) → 0B31F95A9GDAE4H6EG28C781463D
- 21 (37) → 08DB37565F184FA3G0H946EACBC2G9D27E1H
In Base 19:
- 7 (7) → 2DAG58
- B (11) → 1DFA6H538C
- D (13) → 18EBD2HA475G
In Base 20:
- 3 (3) → 6D
- D (13) → 1AF7DGI94C63
- H (17) → 13ABF5HCIG984E27
In Base 21:
- J (19) → 1248HE7F9JIGC36D5B
- 12 (23) → 0J3DECG92FAK1H7684BI5A
- 18 (29) → 0F475198EA2IH7K5GDFJBC6AI23D
In Base 22:
- 5 (5) → 48HD
- H (17) → 16A7GI2CKFBE53J9
- J (17) → 13A95H826KIBCG4DJF
In Base 23:
- 3 (3) → 7F
- 5 (5) → 4DI9
- H (17) → 182G59AILEK6HDC4
In Base 24:
- 7 (7) → 3A6KDH
- B (11) → 248HALJF6D
- D (13) → 1L795CM3GEIB
- H (17) → 19L45FCGME2JI8B7
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.
It can be shown that no cyclic numbers (other than trivial single digits) exist in any numeric base which is a perfect square; thus there are no cyclic numbers in hexadecimal, base 4, or nonary.
See also
References
Further reading
- 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.
- Kalman, Dan; 'Fractions with Cycling Digit Patterns' The College Mathematics Journal, Vol. 27, No. 2. (Mar., 1996), pp. 109-115.
- 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
- Wells, David; "The Penguin Dictionary of Curious and Interesting Numbers", Penguin Press. ISBN 0-14-008029-5