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...
The conjecture is wrong.
The conjecture is wrong.
David BroadHurst has found counter-examples.
The terrible "law of small numbers" has struck again... :cry::mad::confused::frown: (but the numbers were not so small...).
I've updated the paper and just conjectured that the highest power of 3...
Hi,
I have the following (new, I think) conjecture about the Mersenne prime numbers, where: M_q = 2^q - 1 with q prime.
I've checked it up to q = 110503 (M29).
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...
New version of the conjectures !
Hello,
I've produced a http://tony.reix.free.fr/Mersenne/SummaryOfThe3Conjectures.pdf" [Broken] (thanks to "Dodo"), with more explanations, with new seeds, and with new PARI/gp code, and with information about the status of proof of the sufficiency, and who did...
Not only a matter of primality proving. More.
For those who are reading this thread, let me remind that this thread does not only deal with some new theorems helping to prove that some special numbers (Mersenne, Wagstaff, Fermat) are primes.
If it was only this, then it would only worth to...
Yes ! You are perfectly right !
Clear that this problem you are talking about does not appear when using LLT with the Digraph-tree, since once you've reached Si=0 then the next steps are: -2, 2, 2, 2, ...
With cycles, yes there is the risk to be in a (q-1)/m cycle. So that, after q-1...
INdex from 1 to q, instead of 0 to q-1.
Good comment ! It is exactly what Mr Gerbicz did in his proof !
Notice that it is possible to use the same seed 1/4 for Conjectures 2 & 3.
Because all cycles are of length (q-1)/m, I prefer that the seed be named S0. So that the iterations are from 1...
2 proofs of the first part of Conjectures 2 & 3
Hi,
Mister Robert Gerbicz has provided a proof for the first (the easier, but not so easy !) part of Conjectures 2 and 3. See http://robert.gerbicz.googlepages.com/WagstaffAndFermat.pdf" [Broken].
Though this paper needs some cleaning and...
A .pdf summary of conjectures
I've summarized the information in this http://tony.reix.free.fr/Mersenne/SummaryOfThe3Conjectures.pdf" [Broken].
And here is a link to a http://www.cs.uwaterloo.ca/~tmjvasig/papers/newvasiga.pdf" [Broken]r.
Tony
Hi,
I've put on the http://mersenneforum.org/showthread.php?t=10670" the description of 3 conjectures that are waiting for a proof. I've already done half the proof for one of them (the easy part...).
I've provided PARI/gp code that exercises the 3 conjectures.
I'll give 100Euro for the...
Yes, the announce is now available.
Two new Mersenne primes bigger than 10M digits.
Look at http://www.mersenne.org/" [Broken] web-site and come contribute !
There is a lot of fun there !
Tony
Yes: V_{2n}=V_n^2-2Q^n.
When n is even and Q=-1 or 1, we have: V_{2^n}=V_{2^{n-1}}^2-2 which is the LLT basic formula.
Have I found something useful or is it simply another way to look at an old result ? (Probably second one !)
T.
Lucas numbers
I think I have an idea.
It appears that we have: L^m(i) = V_{2^m}(1,-1), where i is the square root of -1, m is greater than 1, and V_n(1,-1) is a Lucas number defined by: V_0=2 , V_1=1, V_{n+1}=V_n+V_{n-1}. Look at "The Little Book of BIGGER primes" by Paulo Ribenboim, 2nd...
Hello,
I'm an amateur: I play with numbers and try to find nice/interesting properties about nice numbers. I have no proof of this one. I like the way Hurkyl did: he gave an hint and then some of the Maths I learnt 30 years ago plus the Number Theory I've learnt these last years come back and I...
LLT numbers and Fermat primes
Now, more difficult, I think.
Can you prove:
\prod_{i=1}^{2^n-1}C_i^{+} \equiv 1 \pmod{F_n}\ \ iff \ \ F_n=2^{2^n}+1 is prime.
T.
Let's say: L(x)=x^2-2 , L^1 = L, L^m = L \circ L^{m-1} = L \circ L \circ L \ldots \circ L.
Where L(x) is the polynomial used in the Lucas-Lehmer Test (LLT) :
S_0=4 \ , \ S_{i+1}=S_i^2-2=L(S_i) \ ; \ M_q \text{ is prime } \Longleftrightarrow \ S_{q-2} \equiv 0 modulo M_q .
We have...
No idea ?
This conjecture is the first step of an attempt to find a primality proof for Mersenne numbers faster than the Lucas-Lehmer test (LLT).
So a proof could help the GIMPS project.
Tony
Hi,
You probably already know the Lucas-Lehmer-Test (LLT) used for proving that a Mersenne number is prime or composite. (See: http://mathworld.wolfram.com/Lucas-LehmerTest.html" [Broken]).
The LLT is based on the properties of the Tree built by x^2-2 modulo a Mersenne number.
Now, here is a...
Look at this http://www.mersenneforum.org/showthread.php?p=5769&mode=threaded". Follow links at bottom.
I have not clear scalability numbers: I've not run prime95 and Glucas on the same multi-CPUs i386 machine yet.
Glucas speeds up the LLT because the FFT has been parallelized: it uses several...
Both GIMPS prime95 and Guillermo Valor Glucas programs use the Lucas-Lehmer-Test (LLT), based on Lucas sequences. This is a very simple test that can be used for some numbers (maybe more but I'm not aware of): Mersenne numbers, Fermat numbers, and k*2^n +/- 1 (LL-Riesel).
The LLT has been...
The LLT, with different details, applies to different kinds of numbers. As an example, I've wrote papers proving that it can be used for Fermat numbers: http://tony.reix.free.fr/Mersenne/PrimalityTest1FermatNumbers.pdf" [Broken]. This was known long time ago (and forgotten). So, the GIMPS...