# Mersenne number is prime implies number is prime

From Number

## Statement

Suppose is a positive integer such that the Mersenne number

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

## Related facts

## 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.