# Mersenne number is prime implies number is prime

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## Statement

Suppose  is a positive integer such that the Mersenne number



is a prime number. Then  itself is a prime number.

## Proof

We prove the statement in its contrapositive form: if  is not prime, then  is not prime. The case  is immediate, so we consider the case , whereby it must be composite.

Given:  where  are (possibly equal, possibly distinct) positive integers greater than 1.

To prove:  is composite.

Proof: Write . We have a polynomial factorization:



Plug in  and get:



Since  are both greater than 1,  and the other factor are both greater than 1, so  is composite.