20090307, 11:21  #1 
Feb 2004
France
2^{2}·229 Posts 
Order of 3 modulo a Mersenne prime
Hi,
I have the following conjecture about the Mersenne prime numbers, where: with 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): where: . With greatest such that , then we have: but no always: . 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 
20090311, 19:42  #3 
Feb 2004
France
2^{2}·229 Posts 
The conjecture is wrong.
The conjecture is wrong.
David BroadHurst has found counterexamples. 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 
20090312, 17:59  #4 
"Nancy"
Aug 2002
Alexandria
4643_{8} Posts 
Out of curiosity: what are the counterexamples?
Alex 
20090312, 19:42  #5  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
5870_{10} Posts 
Quote:


20090313, 03:31  #7 
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
Tony,
May all your future numbers be large! 
20090313, 10:46  #8 
Feb 2004
France
1624_{8} Posts 

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