# Factor of Mersenne Number Mp is of form 2kp + 1

## Contents

## Theorem

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

Then:

- $q = 2 k p + 1$

for some integer $k$.

## Proof

Let $q \mathrel \backslash M_p$.

Then:

- $2^p \equiv 1 \pmod q$

From Integer to Power of Multiple of Order, the multiplicative order of $2 \pmod q$ divides $p$.

By Fermat's Little Theorem, the multiplicative order of $2 \pmod q$ also divides $q - 1$.

Hence:

- $q - 1 = 2 k p$

$\blacksquare$

## Historical Note

This result was known to Pierre de Fermat by $1640$, and it has been suggested that Marin Mersenne made use of it when providing his famous list of Mersenne primes in his *Cogitata Physico-Mathematica* of $1644$.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Proposition $36$: Footnote