Definition:Mersenne Number

A Mersenne number is a natural number of the form $$2^p - 1$$, where $$p$$ is prime.

The number $$2^p - 1$$ is, in this context, often denoted $$M_p$$.

They are named for Marin Mersenne, who wrote a book on the subject and aroused considerable interest in their study.

Factors of Mersenne Numbers
From Factors of Mersenne Numbers, we see that all divisors of $$2^p - 1$$ are of the form $$2 k p + 1$$.

Thus any factors of Mersenne numbers can conveniently be referred to by the value of $$k$$.

Also see
When a Mersenne number is prime, it is called a Mersenne Prime.


 * Also see the Lucas-Lehmer Test.