Mersenne number for prime or Poulet implies prime or Poulet

Statement

Suppose $\text{[math]}$ is a natural number such that:

$\text{[math]}$ .

Consider the $\text{[math]}$ Mersenne number $\text{[math]}$ . Then, we have:

$\text{[math]}$ .

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 $\text{[math]}$ ), is also either a prime number or a Poulet number.

Related facts

Applications

Infinitude of Poulet numbers

Proof

Given: A natural number $\text{[math]}$ such that $\text{[math]}$ .

To prove: $\text{[math]}$ .

Proof: By assumption, $\text{[math]}$ divides $\text{[math]}$ , so there exists an integer $\text{[math]}$ such that $\text{[math]}$ . Thus, $\text{[math]}$ . Thus, we have: