conjecture about mersenne numbers

sascha77 · 2010-01-06, 21:58

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

gd_barnes · 2010-01-07, 07:20

This should be posted in the GIMPS forum.

sascha77 · 2010-01-07, 08:06

ok. thanks.
I will post this in the gimps->math Forum

2010-01-06, 21:58	#1
sascha77 Jan 2010 germany 32₈ 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

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2010-01-07, 07:20	#2
gd_barnes "Gary" May 2007 Overland Park, KS 5·23·103 Posts	This should be posted in the GIMPS forum.

2010-01-07, 08:06	#3
sascha77 Jan 2010 germany 1A₁₆ Posts	ok. thanks. I will post this in the gimps->math Forum