Full reptend prime

Definition

For arbitrary base

A prime $\text{[math]}$ is termed a full reptend prime to base $\text{[math]}$ (where $\text{[math]}$ is a positive integer greater than 1) if the following equivalent conditions are satisfied:

$\text{[math]}$ is a primitive root modulo $\text{[math]}$ .
The base $\text{[math]}$ expansion of $\text{[math]}$ has repeating block of length $\text{[math]}$ .
The number $\text{[math]}$ is a cyclic number in base $\text{[math]}$ .

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 $\text{[math]}$ , there exists a finite number dependent on $\text{[math]}$ such that whether or not $\text{[math]}$ is a full reptend prime mod $\text{[math]}$ depends only on the congruence class of $\text{[math]}$ modulo that finite number. When $\text{[math]}$ itself is an odd prime, this number is $\text{[math]}$ .

Full reptend prime

Contents

Definition

For arbitrary base

Default of base 10

Facts

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