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 is that this sequence contains 37.395..% of the primes.
Read more about this topic: Cyclic Number
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