# 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 $q\mathrm{\mid}{M}_{p}$, then $p\mathrm{\mid}\mathrm{(}q\mathrm{-}\mathrm{1}\mathrm{)}$.

###### Proof.

By definition of $q$, we have ${2}^{p}\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modq)$. 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 $d:=\mathrm{gcd}({2}^{n}-1,{2}^{m}-1)$. Since $d$ is odd, $2$ is a unit in ${\mathbb{Z}}_{d}$ and, since ${2}^{n}\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modd)$ and ${2}^{m}\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modd)$, the order of $2$ divides both $m$ and $n$: it is $1$. Thus $2\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modd)$ 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 |
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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 |