Full reptend prime

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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:

  1. is a primitive root modulo .
  2. The base expansion of has repeating block of length .
  3. 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 .