Mersenne number for prime or Poulet implies prime or Poulet
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.
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: