A number of the form 2^n -1 is called a Mersenne number, and if it happens to be prime it is called a Mersenne prime. Some exceptionally large Mersenne primes have been discovered.
Question : If n is greater than or equal to 1 and 2^n -1 is a prime, prove that n is prime.
Hint. Start with supposing n is not prime. Then use the following factorization, which holds for any x and any positive m.
x^m - 1 = (x - 1) (x^(m-1) + x^(m-2)..... + x^2 + x + 1)