Mersenne numbers, two small results on

This entry presents two simple results on Mersenne numbers, 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 $q\mathrm{\mid }{M}_p$, then $p\mathrm{\mid }\mathrm{(}q\mathrm{-}\mathrm{1}\mathrm{)}$.

Proof.

By definition of $q$, we have $2^p\equiv 1(modq)$. Since $p$ is prime, this implies that $2$ has order $p$ in the multiplicative group $\mathrm{\setminus }{\mathbb{Z}}_q\{0\}$ and, by Lagrange’s Theorem, it divides the order of this group (http://planetmath.org/Group), which is $q-1$. ∎

Theorem.

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

Proof.

Let $d:=\gcd (2^n-1,2^m-1)$. Since $d$ is odd, $2$ is a unit in ${\mathbb{Z}}_d$ and, since $2^n\equiv 1(modd)$ and $2^m\equiv 1(modd)$, the order of $2$ divides both $m$ and $n$: it is $1$. Thus $2\equiv 1(modd)$ and $d=1$. ∎

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 $M_{11}=23\cdot 89$).

Title	Mersenne numbers, two small results on
Canonical name	MersenneNumbersTwoSmallResultsOn
Date of creation	2013-03-22 15:07:53
Last modified on	2013-03-22 15:07:53
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	12
Author	CWoo (3771)
Entry type	Result
Classification	msc 11A41
Related topic	MersenneNumbers