Mersenne Primes

A Mersenne Prime is a prime number of the form 2ⁿ - 1. The first few examples are 3, 7, 31, etc. The number 5 is not a Mersenne Prime because 6 is not an integral power of 2. Likewise, 11 isn't a Mersenne Prime because 12 isn't 2 to the power of some integer.

It's well known that for 2ⁿ - 1 to be a prime number, n itself must be prime. Still, I was interested in proving this for myself, just as I had proven the Pythagorean Theorem for myself. So one night when my mind was still working after I had gone to bed, this is what occurred to me.

Imagine 2ⁿ - 1 expressed in binary. This will simply be a string of n 1's. If n is composite, then the number of 1's can be grouped into x sets of n/x 1's, where x is one of the factors of n, and x > 1.

Let y = n/x. Then each group of 1's is the binary for 2^y - 1. Therefore 2ⁿ - 1 = x(2^y-1), making 2ⁿ - 1 composite.

Therefore, for 2ⁿ - 1 to be a prime number, it is necessary that n be a prime number.