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:

142857 (6 digits)

0588235294117647 (16 digits)

052631578947368421 (18 digits)

0434782608695652173913 (22 digits)

0344827586206896551724137931 (28 digits)

0212765957446808510638297872340425531914893617 (46 digits)

0169491525423728813559322033898305084745762711864406779661 (58 digits)

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

{\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 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 outputted 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 9's. 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 9 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:

01

0011

0001011101

000100111011

000011010111100101

In ternary:

0121

010212

0011202122110201

001102100221120122

0002210102011122200121202111

In octal:

25

1463

0564272135

0215173454106475626043236713

0115220717545336140465103476625570602324416373126743

In duodecimal:

2497

186A35

08579214B36429A7

In Base 24:

3A6LDH

248HAMKF6D

1L795CN3GEJB

19M45FCGNE2KJ8B7

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.

References

^ http://mathworld.wolfram.com/ArtinsConstant.html

External links

Weisstein, Eric W. "Cyclic Number". MathWorld.

[1] ttp://mathworld.wolfram.com/ArtinsConstant.html

[1]