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Old 2010-01-07, 08:13   #1
sascha77
 
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Jan 2010
germany

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Default 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
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Old 2010-01-07, 18:28   #2
maxal
 
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(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.
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Old 2010-01-08, 08:44   #3
sascha77
 
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germany

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Thanks maxal,

Now I see the connection to the Wieferich primes and
I feel confident now that this is hard to solve.
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Old 2010-02-09, 04:58   #4
blob100
 
Jan 2010

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Do you have a proof to this conjecture (without inclding the 2p-1 prime number option)?
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Old 2010-02-09, 05:29   #5
retina
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Quote:
Originally Posted by blob100 View Post
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.
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Old 2010-02-09, 19:17   #6
blob100
 
Jan 2010

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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?
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Old 2010-02-09, 20:31   #7
R.D. Silverman
 
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Quote:
Originally Posted by blob100 View Post
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.
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Old 2010-02-09, 20:47   #8
blob100
 
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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.
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Old 2010-02-10, 07:34   #9
henryzz
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Quote:
Originally Posted by blob100 View Post
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
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Old 2010-02-10, 16:30   #10
blob100
 
Jan 2010

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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..
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Old 2010-04-25, 23:31   #11
sascha77
 
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germany

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Default 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
File Type: pdf mersenne_conj.pdf (88.2 KB, 237 views)

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