Factor of Mersenne Number

Theorems

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

Then the following properties hold:

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

$q = 2 k p + 1$

for some integer $k$.

Thus any factor of a Mersenne number can conveniently be referred to by the value of $k$.

Factor of Mersenne Number $M_p$ equivalent to $1 \pmod p$

$q \equiv 1 \pmod p$

Factor of Mersenne Number equivalent to $\pm 1 \pmod 8$

$q \equiv \pm 1 \pmod 8$