If 2n-1 is prime, then so is n

The goal of this short "footnote" is to prove the following theorem used in the discussion of Mersenne primes.

Theorem.
If for some positive integer n, 2n-1 is prime, then so is n.
Proof.
Let r and s be positive integers, then the polynomial xrs-1 is xs-1 times xs(r-1) + xs(r-2) + ... + xs + 1.  So if n is composite (say r.s with 1 < s < n), then 2n-1 is also composite (because it is divisible by 2s-1).

Notice that we can say more: suppose n > 1. Since x-1 divides xn-1, for the latter to be prime the former must be one. This gives the following.

Corollary.
Let a and n be integers greater than one.  If an-1 is prime, then a is 2 and n is prime.

Usually the first step in factoring numbers of the forms an-1 (where a and n are positive integers) is to factor the polynomial xn-1.  In this proof we just used the most basic of such factorization rules, see [BLSTW88] for some others.

Printed from the PrimePages <primes.utm.edu> © Chris Caldwell.