# Full reptend prime

From Number

## Definition

### For arbitrary base

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

- is a primitive root modulo .
- The base expansion of has repeating block of length .
- The number is a cyclic number in base .

### 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 , there exists a finite number dependent on such that whether or not is a full reptend prime mod depends only on the congruence class of modulo that finite number. When itself is an odd prime, this number is .