Cyclic Number - Construction of Cyclic Numbers

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.