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 2010-01-07, 08:13 #1 sascha77     Jan 2010 germany 2×13 Posts Mersenne Conjecture 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, 18:28 #2 maxal     Feb 2005 22×5×13 Posts (1) implies (2) for all $a$ if and only if $p^2$ does not divide $M_p-1=2^p-2$. In other words, you need to prove either of the two statements: * if $p$ is Wieferich prime then $M_p$ is not prime; * if $M_p$ is prime then $p$ is not a Wieferich prime. It may be hard to prove.
 2010-01-08, 08:44 #3 sascha77     Jan 2010 germany 2·13 Posts Thanks maxal, Now I see the connection to the Wieferich primes and I feel confident now that this is hard to solve.
 2010-02-09, 04:58 #4 blob100   Jan 2010 379 Posts Do you have a proof to this conjecture (without inclding the 2p-1 prime number option)?
2010-02-09, 05:29   #5
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

6,689 Posts

Quote:
 Originally Posted by blob100 Do you have a proof to this conjecture (without inclding the 2p-1 prime number option)?
If the OP had proof then it wouldn't be a conjecture.

 2010-02-09, 19:17 #6 blob100   Jan 2010 17B16 Posts What do you mean by element ( I tought element is a member in a set). Do you have an example for an element of a merssene number?
2010-02-09, 20:31   #7
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

22·1,877 Posts

Quote:
 Originally Posted by blob100 What do you mean by element ( I tought element is a member in a set). Do you have an example for an element of a merssene number?
It is bad nomenclature. He means that a is an element of the group of
units of Z/(2^p-1)Z

, i.e. an element of its maximal cyclic subgroup.

 2010-02-09, 20:47 #8 blob100   Jan 2010 379 Posts Silverman, So he just mean a factor of 2p-1. BTW: I started reading "solved and unsloved problems in mathematics", I really enjoy it and it is really readable for my level.
2010-02-10, 07:34   #9
henryzz
Just call me Henry

"David"
Sep 2007
Liverpool (GMT/BST)

179216 Posts

Quote:
 Originally Posted by blob100 Silverman, So he just mean a factor of 2p-1. BTW: I started reading "solved and unsloved problems in mathematics", I really enjoy it and it is really readable for my level.
Do you mean D. Shanks "Solved and Unsolved Problems in Number Theory"?
Are you learning from it or just enjoying it?
Might get it myself

 2010-02-10, 16:30 #10 blob100   Jan 2010 5738 Posts Henry, I do read it, it isn't easy and it takes me time to understand. When I say "enjoy", I mean that it is really fun for me to learn more mathematics, I love mathematics, so why won't I enjoy reading a math book? I'm not going to read a book if I'm going to be bored of it..
2010-04-25, 23:31   #11
sascha77

Jan 2010
germany

2·13 Posts
Explanation of my conjecture.

Hello ,

I wrote an PDF-document to show why I am interested
that "my" conjecture is true.
Sorry that it took so long time ;-(

In this PDF there might be some "new" Mersenne Properties, but
I am not really shure if they are not well known.
So I post this document here, in hope that the information might
be usefull for someone.
If I did mathematical mistakes in the document I would really be glad if you can say what I did false.
( I know there are plenty of english-grammar-mistakes ;-) Please excuse my bad english)

kind regards,

Sascha
Attached Files
 mersenne_conj.pdf (88.2 KB, 289 views)

Last fiddled with by sascha77 on 2010-04-25 at 23:47

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