# Search results

1. ### Order of 3 modulo a Mersenne prime

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...
2. ### Order of 3 modulo a Mersenne prime

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...
3. ### Order of 3 modulo a Mersenne prime

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...
4. ### Three conjectures looking for a proof ! 100Euro reward !

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...
5. ### Three conjectures looking for a proof ! 100Euro reward !

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...
6. ### Three conjectures looking for a proof ! 100Euro reward !

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...
7. ### Three conjectures looking for a proof ! 100Euro reward !

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...
8. ### Three conjectures looking for a proof ! 100Euro reward !

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...
9. ### Three conjectures looking for a proof ! 100Euro reward !

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
10. ### Three conjectures looking for a proof ! 100Euro reward !

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...
11. ### TWO new Mersenne primes?

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
12. ### Lucas-Lehmer Test With Polynomials

Don't we should have all coefficients lower than x=4 ? As an example, 3x - 4 = 2x ? T.
13. ### LLT numbers

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.
14. ### LLT numbers

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...
15. ### LLT numbers

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...
16. ### LLT numbers

Better, simpler Can you prove: C_{2^n}^{+} \equiv -2 \pmod{F_n} \ \Longleftrightarrow \ F_n=2^{2^n}+1 \text{ is prime.}. T.
17. ### LLT numbers

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.
18. ### LLT numbers

Got it ! OK. I've got a proof. Thanks for the hints! Not so difficult once you think using "i" ! T.
19. ### LLT numbers

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...
20. ### Another (candidate) test of primality of Mersenne number

Since 9 = 2 mod 7 and since 2*4=8=1 mod 7 , 9+1/9 = 2+1/2=2+4=6 mod 7. T.
21. ### Another (candidate) test of primality of Mersenne number

It is Arithmetic. Use an appropriate tool, like PARI/gp. Here y=1/x=x^(-1) mod z means y*x = 1 mod z , where x, y and z are integers .... T.
22. ### Another (candidate) test of primality of Mersenne number

I forgot to say that the idea could only speed up proving a Mersenne number is composite. T.
23. ### Another (candidate) test of primality of Mersenne number

Iterations for q=5,7,11,13 and PARI/gp code q=5: S0=16 -> 6 -> 3 -> 7 -> 16 q=7: S0=122 -> 23-> 19 -> 105 -> 101 -> 39 -> 122 q=11: S0=464 -> 359 -> 1965 -> 581 -> 1851 -> 1568 -> 175 -> 1965 -> 581 -> 1851 -> 1568 q=13: S0=7290 -> 890 -> 5762 -> 2519 -> 5525 -> 5957 -> 2435 -> 7130 ->...
24. ### Another (candidate) test of primality of Mersenne number

1/3^2 mod Mq can be easily computed, since Mq=1+2.3.q.k . 1/3 mod Mq = (2.Mq+1)/3 . Examples with q=5, 7, 13 (using PARI/gp) : q=5 (1/3)%31 --> 21 (2*31+1)/31 --> 21 (3^2+1/3^2)%31 --> 16 q=7 (1/3)%127 --> 85 (2*127+1)/3 --> 85 (3^2+1/3^2)%127 --> 122 q=13...
25. ### Another (candidate) test of primality of Mersenne number

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
26. ### Another (candidate) test of primality of Mersenne number

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...
27. ### M43: GIMPS project has found a new Mersenne prime

Would you mind elaborate ? What is ridiculous ?
28. ### M43: GIMPS project has found a new Mersenne prime

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...
29. ### M43: GIMPS project has found a new Mersenne prime

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...
30. ### M43: GIMPS project has found a new Mersenne prime

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...