20080213, 07:54  #23  
Jan 2008
10110_{2} Posts 
p need to be prime. It is possible to made a generalisation for an unspecified p, but then it add a little bit more complexity on the conjecture.
I don't have put that general case on my paper, because compared with conjecture no 1, conjecture no 2 and 3 are only a very particular but interesting case to study. Conjecture 2 and 3 show clearly that we can go beyond the 3 century old "classical modular restriction" and up to now, almost everyone thought that it would be impossible and desperate to go beyond. For conjecture no 1 it is also possible to made a generalisation for an unspecified p, and so for the Fermat numbers. It is actually one of my tasks. Quote:


20080214, 17:07  #24  
Feb 2007
2^{4}·3^{3} Posts 
Quote:
Thus the conjecture, to be correct, needs to be written differently, namely, the "p prime" must be taken away from the r.h.s. of the "if and only if", and must be put in front of the whole: Let p be a prime. Then xxx iff yyy. 

20080214, 23:13  #25 
Jan 2008
22_{10} Posts 
2nd numerical exemple (conjecture no 3)
Hi everyone!
You will find in attachment a pdf file that contain a numerical exemple for conjecture 3, for the special case j=4 (the set of probable divisors then reduced to the set of all divisors). All the potential divisors are examined up to p=2000000. Conjecture 3 (for j=4 and M): Let p be prime, d=2*p*j+1=2*p*4+1 divide M(p)=2^p1 if and only if
Olivier 
20080215, 01:27  #26 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
17BB_{16} Posts 
So now what? With your numerology you have some numbers that "work" up to 2000000. Are you going to formalise it with some actual mathematics?

20080215, 08:00  #27 
Jan 2008
26_{8} Posts 
I have "stop" at p=2000000 for the two exemples because I was limited for the size of the pdf file that I can upload on the forum.
In fact, I have test these two conjectures for values of p much greater (up to 10^11), but of course I was limited by the power of the computer that I have at my disposition (a 56 CPU SGI) to go beyond. I have made also other tests for j greater than 3 for conjecture 2, and j greater than 4 for conjecture 3. I have made also plenty of other tests namely for M+(p)=2^p+1. Of course, so far, I was unable to found any counter exemple. I hope I will not have to wait 3 century to see the demonstrations ...but I'm a little pessimistic about that because I think (and not only me) that it will be complicated. Now, I'm very sorry if I don't use the "classical" terms of formulation of mathematics, I recognize that is one of my weakeness. 
20080215, 09:18  #28 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3^{5}×5^{2} Posts 

20080217, 22:10  #29  
Jan 2008
2×11 Posts 
Quote:
The cubic reciprocity that R.D. Silvermann spoke in thread 2 is only the “other side of the mirror”, it is an equivalence but these prohibited not the possible existence of something else that could say when the cubic residue is automatically verified. When I started this work almost 10 years ago, I was convinced that it was possible to go beyond the “classical” modularity’s restrictions, and I succeeded, it was a bet without a guaranty of success. For now, we don’t have a demonstration for any of the three conjectures of the paper, but all these conjectures have been verified to a very wide range of parameters. Unlike some other conjectures (like for instance the famous Goldbach conjecture) we can easily determine and increase the level of “verification” of the conjecture. I explain me: for j=3 the probability that a candidate divisor (p prime, mod(d,8)=7 and d prime) is a divisor of M(p)=2^p1 is 1/3. In the example I gave (see thread 20) there is 13082 candidate divisors and 4315 divisors, so the probability that conjecture 2 give by chance all divisors for p up to 2x10^6 is about of 1/3^4315 …which is very very tiny. And I have been much further (p up to 10^11) for j=3 for M+ and M. If you think that conjecture 2 is not true you should then find easily a counterexample otherwise I’m the luckiest man of the universe and I should buy a lottery ticket! So, like Fivemack says in thread 12, it is pretty sure that conjecture 2 is true. The other conjectures (3 and 1) have also been verified to a tremendous level, thanks to a big 56 CPU SGI computer that I have at disposition. You should also not forget that conjecture 1 is the general case for any value of j, any p prime and for any base b prime of the Cunningham numbers C+=b^p+1 which is the reason why the title of my work is “Towards an ultimate and unified theory of factorization of Cunningham numbers”; So, the ideas behind my work are very strong, although it is always possible to change the presentation to avoid heartbreaking to some peoples. In a few days I will post a new discussion about the square free conjecture analyzed with conjecture 1. 

20080218, 21:53  #30  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Number theory is Dr. Silverman's profession.
You are, in this thread, discussing number theory, not physics. It is Dr. Silverman, not you, who is the greater authority in the field of this current discussion. Your dismissals of Dr. Silverman's responses (posts #2, 4, and 6 of this thread) betray your ignorance rather than your competence. Quote:
Quote:
However, in mathematics the believability of your work is more related to its demonstrated logical consistency with known proven theorems than to any level of "impossibility". (See http://en.wikipedia.org/wiki/BanachTarski_paradox and http://www.kuro5hin.org/story/2003/5/23/134430/275 for an example.) Until you demonstrate that you understand what Dr. Silverman wrote to you, no knowlegable participant in this forum will have much respect for your unproven statements (not to mention your justplainfalse statements). Quote:
Quote:
Quote:
Last fiddled with by cheesehead on 20080218 at 21:56 

20080219, 23:17  #31 
Jan 2008
2·11 Posts 
Dear Cheesehead,
There is a general principle well above physics, chemistry, biology … and mathematics too: theory need to take into account observations (in some Mersenne papers of H.W. Lenstra there is this concept of “observations”). You can see the three conjectures of my paper http://olivier.latinne.googlepages.com/, as a set of observations. See for instance thread 20 (example for conjecture 2) or thread 25 (example for conjecture 3). You can’t negate strong observations especially when other people (like Fivemack, see thread 12) verify independently your claim. All three conjectures have been verified for billions of candidate divisors without a single counterexample. Now, to find demonstrations it’s an other story … and I will not venture. I know perfectly my limits and I will leave this task to mathematicians, because I think that it will be a very difficult task. Please, let me recall you what P. Deligne who is a Fields Medals winner of 1978 say to me (from a private communication): “ … the (non) divisibility by small prime numbers could destroy the HardyLittlewood heuristic …”. Clearly, he worry that additional restrictions to the “classical” modular restrictions could destroy the HardyLittlewood heuristic. But he never says that there is an impossibility and he never says anything about the j reciprocity law that is the only possible criteria and that nothing else could exist. Best regards, Olivier 
20080220, 00:50  #32 
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Olivier,
Perhaps there is a simple language confusion here that I should have addressed first. I am puzzled by your use of "demonstration". I thought you were using it in the sense of "showing specific examples" (and I wondered how you meant that to differ from "verification"). Do you instead mean it to have the sense of "logical proof"? Or ... ? I was addressing my preceding comments to those of yours where you wrote "demonstration" but never used the word "observation". Last fiddled with by cheesehead on 20080220 at 00:58 
20080220, 10:34  #33  
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Quote:
Oh come let us adore him Oh come let us adore him Christ the Lord. 

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