Mersenne number for prime or Poulet implies prime or Poulet

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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.

Related facts

Applications

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:

.