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 2010-01-06, 21:58 #1 sascha77     Jan 2010 germany 2·13 Posts conjecture about mersenne numbers Hello, My conjecture is : Let $2^p-1$ be an Mersenne-number, and a is an element from $2^p-1$ $(1)\:\: a^{(p*p)} \equiv 1\: (mod\: 2^p-1)$ $(2)\:\: a^{p} \equiv 1\: (mod \: 2^p-1)$ (1) --> (2) This means, that if (1) is true, than is (2) also true. It is easy to show that ? My idea to do this with $2^p-1$ is prime, was the following: $a^{pp} \equiv 1$ $a^{p} \equiv \sqrt[p]{1}$ When $2^p-1$ is prime, when the Elements with the form $2^x$ are the only ones, that have order of $p$ and therefore: $a^p \equiv \sqrt[p]{1} \equiv 2^x$ $a \equiv \sqrt[p]{2^x}$ - But the only solution to this is 1: --> $a \equiv \sqrt[p]{2^x}\equiv 1$ $a \equiv 1$ -> So $pp$ can not be the order of a, because a is 1. I have searched with google many sites, but could not find the answer to this "problem". I hope that anybody can help me with the conjecture. kind regards, sascha
 2010-01-07, 07:20 #2 gd_barnes     May 2007 Kansas; USA 2×3×72×37 Posts This should be posted in the GIMPS forum.
 2010-01-07, 08:06 #3 sascha77     Jan 2010 germany 2·13 Posts ok. thanks. I will post this in the gimps->math Forum

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