mersenneforum.org > Math Order of 3 modulo a Mersenne prime
 Register FAQ Search Today's Posts Mark Forums Read

 2009-03-07, 11:21 #1 T.Rex     Feb 2004 France 2×33×17 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 39616 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 2·33·17 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 46438 Posts Out of curiosity: what are the counter-examples? Alex
2009-03-12, 19:42   #5
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

172416 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 91810 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

2·33·17 Posts

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

 Similar Threads Thread Thread Starter Forum Replies Last Post carpetpool Miscellaneous Math 2 2018-04-15 00:28 axn Computer Science & Computational Number Theory 66 2011-09-01 21:55 Raman Math 1 2010-02-16 21:25 T.Rex Math 9 2007-03-26 17:35 Dresdenboy Programming 10 2004-02-29 17:27

All times are UTC. The time now is 17:17.

Wed Oct 27 17:17:08 UTC 2021 up 96 days, 11:46, 0 users, load averages: 1.40, 1.26, 1.21