# Mersenne number for prime or Poulet implies prime or Poulet

## Statement

Suppose  is a natural number such that:

.

Consider the  Mersenne number . Then, we have:

.

In other words, the Mersenne number for a number that is either a prime number or a Poulet number (i.e., an odd composite number that is pseudoprime to base ), is also either a prime number or a Poulet number.

## Proof

Given: A natural number  such that .

To prove: .

Proof: By assumption,  divides , so there exists an integer  such that . Thus, . Thus, we have:

.

Since  divides , we get that:

.