full reptend prime

If for a prime numberMathworldPlanetmath p in a given base b such that gcd(p,b)=1, the formula


gives a cyclic number, then p is a full reptend primeMathworldPlanetmath or long prime.

The first few base 10 full reptend primes are given by A001913 of Sloane’s OEIS: 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167.

For example, the case b=10, p=7 gives the cyclic number 142857, thus, 7 is a full reptend prime.

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 theoryMathworldPlanetmath, specifically, this sequence is the set of primes p such that 10 is a primitive rootMathworldPlanetmath modulo p. A conjecture of Emil Artin on primitive roots is that this sequence contains about 37 percent 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.”

Title full reptend prime
Canonical name FullReptendPrime
Date of creation 2013-03-22 16:04:50
Last modified on 2013-03-22 16:04:50
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11N05
Synonym long prime