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 2009-03-07, 11:21 #1 T.Rex     Feb 2004 France 3×311 Posts 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
 2009-03-08, 09:25 #2 T.Rex     Feb 2004 France 3·311 Posts Hi, The following theorem, dealing still with the order of 3 modulo a Mersenne prime, proved by ZetaX, could help: Theorem. Tony
 2009-03-11, 19:42 #3 T.Rex     Feb 2004 France 3×311 Posts 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
 2009-03-12, 17:59 #4 akruppa     "Nancy" Aug 2002 Alexandria 1001101000112 Posts Out of curiosity: what are the counter-examples? Alex
2009-03-12, 19:42   #5
henryzz
Just call me Henry

"David"
Sep 2007
Liverpool (GMT/BST)

136178 Posts

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

 2009-03-12, 20:00 #6 T.Rex     Feb 2004 France 3×311 Posts I've updated the paper. The terrible "law of small numbers"... Tony
 2009-03-13, 03:31 #7 cheesehead     "Richard B. Woods" Aug 2002 Wisconsin USA 22·3·641 Posts Tony, May all your future numbers be large!
2009-03-13, 10:46   #8
T.Rex

Feb 2004
France

3·311 Posts

Quote:
Yes ! Thanks ! I'm looking for a big PRP... but it hides behind the Moon...
Tony

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