New Mersenne and Cunningham conjecture

[olivier_latinne]

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.

Originally Posted by m_f_h

I don't understand why you leave j here :
this is supposed to be true only for j=3 ?!
Since e.g. 2^20-1 is a counter example (d=2*20*1+1 divides, but p=20 is not prime)

So it should be rewritten:

d=6p+1 divides M(p)=2^p-1 if and only if

• d=6p+1 is prime
• mod(d,8)=7 [or, a bit simpler (subtract 1 and divide by 2):
  3p=3 (mod 4), i.e.: p=1 (mod 4)]
• p prime
• and there exist integers n and i such that: d=4*n^2 + 3*(3+6*i)^2

But this is clearly wrong:

p=21 is not prime, but 6p+1=127 divides Mp=2097151
p=72 is not prime, but 6p+1=433 divides Mp=4722366482869645213695
p=72 != 1 (mod 4), but 6p+1=433 divides Mp=4722366482869645213695
p=76 is not prime, but 6p+1=457 divides Mp=75557863725914323419135
p=76 != 1 (mod 4), but 6p+1=457 divides Mp=75557863725914323419135

???

[m_f_h]

Originally Posted by retina

m_f_h: I thought p was meant to be prime? So your examples where p != prime would seem to be not relevant?

Yes it seems as if.
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.

[olivier_latinne]

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^p-1 if and only if

• d is prime
• mod(d,8)=1
• and there exists integer n and i such that: d=n^2 + (8+16*i)^2

Best Regards,

Olivier

[retina]

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?

[olivier_latinne]

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.

Originally Posted by retina

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?

[retina]

Originally Posted by olivier_latinne

I hope I will not have to wait 3 century to see the demonstrations

Try reading the very first reply in this thread, then you won't have to wait "3 century".

[olivier_latinne]

Originally Posted by retina

Try reading the very first reply in this thread, then you won't have to wait "3 century".

Clearly you seem to be doubtful about my work. Well, it will not be easy but I will try to convince you! Research in physics is my profession and I had found in the past several very new things (like for instance my work on theoretical quantum physics, published in Physical Review Letters: http://prola.aps.org/abstract/PRL/v74/i1/p46_1). I know by these experiences that the level that people believed at your work is inversely proportional to the level of “impossibility” of what you have found. Like I have mention in the introduction of my paper about the factorization of Cunningham numbers (http://olivier.latinne.googlepages.com/home) nothing fundamentally new (about the factors), except the square free conjecture, has been found during these last three century. Moreover, almost everyone up to now thought that it would be impossible to go beyond the “classical” modularity’s restrictions. For instance P. Deligne in a private communication says that (translation from French) “ … the (non) divisibility by small prime numbers could destroy the Hardy-Littlewood heuristic and even if we can be convinced that these arguments should be legitimate, due to the fact that these numbers are so sparsely distributed, it is desperate with the current knowledge to pass from heuristic to proves”.
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^p-1 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 counter-example 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.

[cheesehead]

Originally Posted by olivier_latinne

Research in physics is my profession

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.

and I had found in the past several very new things (like for instance my work on theoretical quantum physics, published in Physical Review Letters: http://prola.aps.org/abstract/PRL/v74/i1/p46_1).

Number theory conjectures are more subject to logical proof than are quantum physics conjectures.

I know by these experiences that the level that people believed at your work is inversely proportional to the level of “impossibility” of what you have found.

... in theoretical quantum physics, perhaps.


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/Banach-Tarski_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 just-plain-false statements).

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.

If you can actually prove that assertion, please do so. If you can't, please respect what Dr. Silverman wrote.

For now, we don’t have a demonstration for any of the three conjectures of the paper,

Absence of demonstration may not be a killer obstacle in theoretical quantum physics ... but here you're discussing mathematics, where the rules of evidence differ.

In a few days I will post a new discussion about the square free conjecture analyzed with conjecture 1.

We hope it will use standard mathematical language and notation, and demonstrate logical consistency with known previously-proven theorems.

[olivier_latinne]

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 counter-example.
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 Hardy-Littlewood heuristic …”. Clearly, he worry that additional restrictions to the “classical” modular restrictions could destroy the Hardy-Littlewood 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

[cheesehead]

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 ... ?

Originally Posted by olivier_latinne

need to take into account observations (in some Mersenne papers of H.W. Lenstra there is this concept of “observations”).

I was addressing my preceding comments to those of yours where you wrote "demonstration" but never used the word "observation".

[davieddy]

Originally Posted by cheesehead

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.

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Christ the Lord.