Full reptend prime

Definition

For arbitrary base

A prime $p$ is termed a full reptend prime to base $b$ (where $b$ is a positive integer greater than 1) if the following equivalent conditions are satisfied:

1. $b$ is a primitive root modulo $p$.
2. The base $b$ expansion of $1/p$ has repeating block of length $p-1$.
3. The number $(b^{p-1}-1)/p$ is a cyclic number in base $b$.

Default of base 10

By default, the term full reptend prime is used for a prime that is a full reptend prime in base 10.

Facts

For a fixed base $b$, there exists a finite number dependent on $b$ such that whether or not $p$ is a full reptend prime mod $b$ depends only on the congruence class of $p$ modulo that finite number. When $b$ itself is an odd prime, this number is $4b$.