## You are here

HomeMersenne numbers, two small results on

## Primary tabs

# Mersenne numbers, two small results on

This entry presents two simple results on Mersenne numbers^{1}^{1}In this entry, the Mersenne numbers are indexed by the primes., namely that any two Mersenne numbers are relatively prime and that any prime dividing a Mersenne number $M_{p}$ is greater than $p$. We prove something slightly stronger for both these results:

###### Theorem.

If $q$ is a prime such that $q\!\mid\!M_{p}$, then $p\!\mid\!(q-1)$.

###### Proof.

By definition of $q$, we have $2^{p}\equiv 1\;\;(\mathop{{\rm mod}}q)$. Since $p$ is prime, this implies that $2$ has order $p$ in the multiplicative group $\mathbb{Z}_{q}\mathbin{\setminus}\{0\}$ and, by Lagrange’s Theorem, it divides the order of this group, which is $q-1$. ∎

###### Theorem.

If $m$ and $n$ are relatively prime positive integers, then $2^{m}-1$ and $2^{n}-1$ are also relatively prime.

###### Proof.

Note that these two facts can be easily converted into proofs of the infinity of primes: indeed, the first one constructs a prime bigger than any prime $p$ and the second easily implies that, if there were finitely many primes, every $M_{p}$ (since there would be as many Mersenne numbers as primes) is a prime power, which is clearly false (consider $M_{{11}}=23\cdot 89$).

## Mathematics Subject Classification

11A41*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections