Order of 3 modulo a Mersenne prime

T.Rex · 2009-03-07, 11:21

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

T.Rex · 2009-03-08, 09:25

Hi,

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

Tony

T.Rex · 2009-03-11, 19:42

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

akruppa · 2009-03-12, 17:59

Out of curiosity: what are the counter-examples?

Alex

henryzz · 2009-03-12, 19:42

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

T.Rex · 2009-03-12, 20:00

I've updated the paper.
The terrible "law of small numbers"...

Tony

cheesehead · 2009-03-13, 03:31

Tony,

May all your future numbers be large!

T.Rex · 2009-03-13, 10:46

Quote:

Originally Posted by cheesehead

May all your future numbers be large!

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

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

2009-03-11, 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

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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

2009-03-12, 17:59	#4
akruppa "Nancy" Aug 2002 Alexandria 100110100011₂ Posts	Out of curiosity: what are the counter-examples? Alex

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!