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Old 2009-03-07, 11:21   #1
T.Rex
 
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Feb 2004
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Default Order of 3 modulo a Mersenne prime

Hi,

I have the following conjecture about the Mersenne prime numbers, where: M_q = 2^q - 1 with q prime.
I've checked it up to q = 110503 (M29).
I previously talked of this conjecture on Mersenne forum ; but I have now more experimental data.

Conjecture (Reix): \large \ order(3,M_q) = \frac {M_q - 1}{3^O} where: \ \large O = 0,1,2 .

With I = greatest i such that M_q \equiv 1 \ \pmod{3^i} , then we have: O \leq I but no always: O = I .

A longer description with experimental data is available at: ConjectureOrder3Mersenne.

Samuel Wagstaff was not aware of this conjecture and has no idea (yet) about how to prove it.

I need a proof...
Any idea ?

Tony
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Old 2009-03-08, 09:25   #2
T.Rex
 
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Hi,

The following theorem, dealing still with the order of 3 modulo a Mersenne prime, proved by ZetaX, could help: Theorem.

Tony
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Old 2009-03-11, 19:42   #3
T.Rex
 
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Default The conjecture is wrong.

The conjecture is wrong.
David BroadHurst has found counter-examples.
The "law of small numbers" has struck again... (but the numbers were not so small...).
I've updated the paper and just conjectured that the highest power of 3 that divides the order of 3 mod M_q is 2. But it is not so much interesting...
Never mind, we learn by knowing what's false too.
Tony
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Old 2009-03-12, 17:59   #4
akruppa
 
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Aug 2002
Alexandria

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Out of curiosity: what are the counter-examples?

Alex
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Old 2009-03-12, 19:42   #5
henryzz
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"David"
Sep 2007
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Default

Quote:
Originally Posted by David Broadhurst on primeform yahoo group
Here is a concise summary:

Tony conjectured that if M = 2^q - 1 is prime for q > 2,
then there exists no prime p > 3 that divides
(M-1)/znorder(Mod(3,M)).

Here are my 10 counterexamples:

[ q, p]

[ 3217, 13]
[ 9689, 29]
[ 9941, 5]
[ 11213, 5]
[ 23209, 5]
[ 44497, 7]
[110503, 7]
[132049, 5]
[132049, 7]
[216091, 71]

Any advance on 10?

David
this is all the counter examples he found
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Old 2009-03-12, 20:00   #6
T.Rex
 
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I've updated the paper.
The terrible "law of small numbers"...
Tony
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Old 2009-03-13, 03:31   #7
cheesehead
 
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"Richard B. Woods"
Aug 2002
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Tony,

May all your future numbers be large!
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Old 2009-03-13, 10:46   #8
T.Rex
 
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Quote:
Originally Posted by cheesehead View Post
May all your future numbers be large!
Yes ! Thanks ! I'm looking for a big PRP... but it hides behind the Moon...
Tony
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