Cyclic number

A cyclic number is an integer for which cyclic permutations 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

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:
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:

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
Conversely, if the digital period of 1/p is
then the digits represent a cyclic number.
For example:
Multiples of these fractions exhibit cyclic permutation:

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
where b is the number base, and p is a prime that does not divide 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.
The first values of p for which this formula produces cyclic numbers in decimal are
For b = 12, these ps are
For b = 2, these ps are
For b = 3, these ps are
There are no such ps in the hexadecimal system.
The known pattern to this sequence comes from algebraic number theory, 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.

Construction of cyclic numbers

Cyclic numbers can be constructed by the following procedure:
Let b be the number base
Let p be a prime that does not divide b.
Let t = 0.
Let r = 1.
Let n = 0.
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
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. For example, in decimal, 142857 × 7 = 999999.
When split into groups of equal length, 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 and the sum of the remainder is a multiple of the divisor.

Other numeric bases

Using the above technique, cyclic numbers can be found in other numeric bases. 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:
In ternary:
In quaternary, there are none.
In quinary:
In senary:
In base 7:
In octal:
In nonary, the unique cyclic number is
In base 11:
In duodecimal:
In ternary, 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 exist in any numeric base which is a perfect square, that is, base 4, 9, 16, 25, etc.