Cyclic number

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A cyclic number is an integer whose cyclic rotations of its digits are consecutive multiples of the number. The best-known decimal example is 142857. Its first six multiples are 142857, 285714, 428571, 571428, 714285, and 857142—all rotations of the original number.

Note on leading zeros: if you disallow leading zeros on numbers, 142857 is the only decimal cyclic number. If you allow leading zeros, more examples can appear, but they follow a more complex rule.

Why they exist: cyclic numbers come from fractions. In base b, a prime p that does not divide b can produce a cyclic number of length p−1 when the repeating digits of 1/p in base b form a full cycle. Such primes are called full reptend primes (or long primes) for base b. The pattern is tied to whether b is a primitive root modulo p.

Examples:
- In base 10, p = 7 gives the cyclic number 142857.
- In base 12, p = 5 gives the cyclic number 2497.

Not all primes work: for base 10, p = 13 does not produce a cyclic number, even though you can write 076923076923 as a repeating block from 1/13.

How to construct one: choose a base b and a prime p that does not divide b. Write 1/p in base b using long division. If the repeating block has length p−1, the digits form a cyclic number of length p−1. If the period is shorter, the result is not a cyclic number.

In short: cyclic numbers are special digit strings whose rotations mirror multiplying by consecutive integers, with the 142857 example in base 10 serving as the classic case. They arise from the decimal (or base-b) expansion of unit fractions 1/p for primes p with a full reptend property.