Mersenne number for prime or Poulet implies prime or Poulet

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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



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: