Register to reply

Order of 3 modulo a Mersenne prime

by T.Rex
Tags: mersenne, modulo, order, prime
Share this thread:
T.Rex
#1
Mar7-09, 05:45 AM
P: 62
Hi,

I have the following (new, I think) conjecture about the Mersenne prime numbers, where: [tex]M_q = 2^q - 1[/tex] with [tex]q[/tex] prime.
I've checked it up to q = 110503 (M29).

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

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

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
Phys.Org News Partner Science news on Phys.org
Wildfires and other burns play bigger role in climate change, professor finds
SR Labs research to expose BadUSB next week in Vegas
New study advances 'DNA revolution,' tells butterflies' evolutionary history
robert Ihnot
#2
Mar14-09, 04:39 PM
PF Gold
P: 1,059
If I understand this correctly we are supposing that 3^3 is the highest dividing power, but take the 27th Mersenne prime, as shown in a table, and consider: [tex]\frac{2^{44496}-1}{81} [/tex] is an integer.

Also, I would suggest trying to check out the 40th Mersenne prime, and find, [tex]\frac{2^{20996010}-1}{243}[/tex] is an integer.
robert Ihnot
#3
Mar14-09, 06:22 PM
PF Gold
P: 1,059
T.Rex: I've checked it up to q = 110503 (M29)

If you want to see some check work on Mersenne 27, notice that 2^2000==4 Mod 81.

Thus dividing out 44496/2000 = 22 + Remainder 496. 496 = 2*248. Thus:

[tex]4^{22}*4^{248}-1\equiv 4^{270}-1 \equiv0 Mod 81 [/tex]

T.Rex
#4
Mar15-09, 02:23 AM
P: 62
Order of 3 modulo a Mersenne prime

The conjecture is wrong.
David BroadHurst has found counter-examples.
The terrible "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.
I've updated the paper.
Sorry, the way David found the counter-examples was not so difficult...
Tony
T.Rex
#5
Mar15-09, 02:31 AM
P: 62
Quote Quote by robert Ihnot View Post
If I understand this correctly we are supposing that 3^3 is the highest dividing power...
Not exactly, Robert. For q=44497, 4 is the highest power of 3 that divides Mq-1, but 1 is the highest power of 3 in the relationship between (Mq-1) and order(3,Mq).
I have other reasons to think that 2 is the highest power of 3 in this relationship. But I need to clarify that before conjecturing again (one mistake is enough !!).
Thanks,
Tony


Register to reply

Related Discussions
Modulo a prime Calculus & Beyond Homework 1
Binomial coefficient modulo a prime Linear & Abstract Algebra 1
Solving polynomial congruences modulo a prime power Calculus & Beyond Homework 0
M43: GIMPS project has found a new Mersenne prime Linear & Abstract Algebra 15
The GIMPS project has found a new Mersenne prime number: M42. Linear & Abstract Algebra 4