Full Reptend Prime

In number theory, a full reptend prime or long prime in base b is a prime number p such that the formula

(where p does not divide b) gives a cyclic number. Therefore the digital expansion of in base b repeats the digits of the corresponding cyclic number infinitely. Base 10 may be assumed if no base is specified.

The first few 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 …

For example, the case b = 10, p = 7 gives the cyclic number 142857, thus, 7 is a full reptend prime. Furthermore, 1 divided by 7 written out in base 10 is 0.142857142857142857142857...

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 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. Artin's conjecture on primitive roots is that this sequence contains 37.395..% of the primes.

The term "long prime" was used by John Conway and Richard Guy in their Book of Numbers. Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers."

The corresponding cyclic number to prime p will possess p - 1 digits if and only if p is a full reptend prime.