Modular restrictions on Mersenne divisors

On this page we prove the following theorem (which we use on our page about Mersenne primesand the historical note "the Largest Known Prime by Year").  Fermat discovered and use the first part of this theorem (p ≡ 1 modulo q) and Euler discovered the second.

Theorem.
Let p and q be odd primes. If p divides Mq, then p ≡ 1 (mod q) and p ≡ +/-1 (mod 8).

Below we give a proof and an example.

Proof.

If p divides Mq, then 2q ≡ 1 (mod p) and the order of 2 (mod p) divides the prime q, so it must be q.  By Fermat's Little Theorem the order of 2 also divides p-1, so p-1 = 2kq.  This gives

2(p-1)/2 = 2qk ≡ 1k ≡ 1 (mod p)

so 2 is a quadratic residue mod p and it follows p ≡ +/-1 (mod 8), which completes the proof.

Example: Suppose p divides M31, then the two parts of the theorem together show p ≡ 1 or 63 (mod 248). By 1772 Euler had used this to show M31 was prime.  

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