Factor of Mersenne Number Mp equivalent to 1 mod p

Theorem

Let $p$ and $q$ be prime numbers such that $q$ is a divisor of the Mersenne number $M_p$.

Then:

$q \equiv 1 \pmod p$

Proof

Let $q \divides M_p$.

From Factor of Mersenne Number $M_p$ is of form $2 k p + 1$:

$q = 2 k p + 1$

and so by definition of congruence modulo an integer:

$q \equiv 1 \pmod p$

immediately.

$\blacksquare$

Sources

• Proof courtesy of The Prime Pages: Modular restrictions on Mersenne divisors.