full reptend prime

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

$$\frac{b^{p-1}-1}{p}$$

gives a cyclic number, then $p$ is a full reptend prime 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 theory, specifically, this sequence is the set of primes $p$ such that 10 is a primitive root 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.”