Mersenne number is prime implies number is prime

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Suppose is a positive integer such that the Mersenne number

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

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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.