New Mersenne and Cunningham conjecture

[cheesehead]

Originally Posted by davieddy

Oh come let us adore him
Oh come let us adore him
Oh come let us adore him
Christ the Lord.

You must not be familiar with my previous postings concerning Silverman!

[davieddy]

Originally Posted by cheesehead

You must not be familiar with my previous postings concerning Silverman!

Point me to some. Was your sycophancy perchance sarcastic?

David

[cheesehead]

Originally Posted by davieddy

Was your sycophancy perchance sarcastic?

You have misinterpreted my Feb. 18 posting; it was neither sycophantic nor sarcastic, but was simply sincere.

I have always respected Dr. Silverman's mathematical prowess. He has numerous published papers in the field of number theory.

However, until a year or two ago I frequently criticized Silverman for his apparent impatience or intolerance toward inexperienced or unknowledgable folks who posted sincere but ignorantly- or awkwardly-worded questions about math here at mersenneforum.org. He seemed (to me) to regard this forum as an extension of the college classrooms in which he was professor, and to expect that forum posters should satisfy the same requirements as students in one of those college classes.

I repeatedly pointed out to Silverman that this public forum had fundamental differences from his classrooms, that it was unreasonable to lambast forum posters for not having the math prerequisites of his classroom students, that his own participation here was completely voluntary on his part rather than part of his job or other assignment, and that he himself often failed to do the same type of preliminary "homework" that he expected the forum questioners to have done.

Originally Posted by davieddy

Point me to some.

Do your own forum search for posts by "cheesehead" containing the keywords "Silverman", "classroom", or "homework", dated at least a year ago. Examine those that I posted shortly after a Silverman posting in the same thread.

[edit:]
Okay, because I'm a softie, here's a good thread with several participants besides me commenting on the Dr.'s style: "Decorum in the forum" at http://mersenneforum.org/showthread.php?t=3946

Also related is "Proposal: 'Math for beginners' subforum" at http://mersenneforum.org/showthread.php?t=3843

[jasong]

Dr. Silverman is a sophomore in the original sense of the word.

Period.

[olivier_latinne]

The discussion of the square free conjecture will be split into two parts. The first one will introduce conjecture 1, with a numerical example in attachment.

Conjectures 2 and 3 are restricted to mod(j,3) = 0 and mod(j,4) = 0 respectively, and are only deterministic for j=3 (see thread 20), j=4 (see thread 25) and j=3*4=12 (which is a combination case of conjecture 2 and 3).

Conjecture 1 (historically found later) is the general deterministic case for any j, any exponent p prime and any base b prime for the prime divisors of C+- = b^p +- 1

Let restricted to base b=2, p prime and M- (for the general case see http://Olivier.Latinne.googlepages.com/home),

d=2*p*j+1 prime divide M- = 2^p-1 if and only if there exists integers x, y, m and k, with x and y coprime, m and j coprime, mod(x,b) ≠ 0 and mod(m,p) ≠ 0, such that:


for j odd: d= (x^j + 2^m * y^j) / k and mod(d,8)=7

for j even: d= (-x^(2*j) + 2^m * y^(2*j) ) / k and mod(d,8)=1



In attachment, I give an example for j=5, for which I have computed all numbers such that:

d= (x^5 + 2 * y^5) / k is prime, mod(d,8)=7, d=2*p*5+1 => p=(d-1)/10 is prime.
(I have choose m=1 and limited 0<x<=2000, 0<y<=2000, 1<=k<=100000)

All these numbers d generated by this way (see attachment) are automatically divisors of M-=2^p-1

[m_f_h]

Olivier: could you show how your restrictions could be useful, e.g. to speed up the search for Mersenne primes ?
For example, to "re-discover" the known Mersenne primes ; more precisely:
For which primes p, for which mprime would start (with normal parameter setting) an LL-test, could you have foreseen that it is composite ?

Or for some primes p which are currently under investigation, could you say "easily" (even if it takes several hours or days) that Mp is not prime ?

Or something else of that kind.

There is no need to go up to 10^11, only about p=40 000 000.
But since you said you have tested the conjecture for all p<10^11, didn't give this to you new results about Mersenne primes beyond what is known?

Anyway, you should point out what is NEW in your statement (I mean in the statement of your conjecture) : Just ONE case for which your conjecture says something more in addition to what
classical results give (in particular, e.g., the criterion explicitely given by R.D.S. in his first reply, using cubic reciprocity).
You always say "it was thought one cannot go beyond..." so you should point out WHERE *you* go beyond!


PS (but this is less important): rather than posting tons of numbers in PDF files, I'd suggest to put the code (or algorithm, if you don't use PARI or Maple or so where it should be not much more than 2-3 lines) used to produce these (this is more meaningful and easier to understand) ; and point out some significant examples and counter-examples.

[davieddy]

Originally Posted by cheesehead

You have misinterpreted my Feb. 18 posting; it was neither sycophantic nor sarcastic, but was simply sincere.

I have always respected Dr. Silverman's mathematical prowess. He has numerous published papers in the field of number theory.

However, until a year or two ago I frequently criticized Silverman for his apparent impatience or intolerance toward inexperienced or unknowledgable folks who posted sincere but ignorantly- or awkwardly-worded questions about math here at mersenneforum.org. He seemed (to me) to regard this forum as an extension of the college classrooms in which he was professor, and to expect that forum posters should satisfy the same requirements as students in one of those college classes.

I repeatedly pointed out to Silverman that this public forum had fundamental differences from his classrooms, that it was unreasonable to lambast forum posters for not having the math prerequisites of his classroom students, that his own participation here was completely voluntary on his part rather than part of his job or other assignment, and that he himself often failed to do the same type of preliminary "homework" that he expected the forum questioners to have done.

Do your own forum search for posts by "cheesehead" containing the keywords "Silverman", "classroom", or "homework", dated at least a year ago. Examine those that I posted shortly after a Silverman posting in the same thread.

[edit:]
Okay, because I'm a softie, here's a good thread with several participants besides me commenting on the Dr.'s style: "Decorum in the forum" at http://mersenneforum.org/showthread.php?t=3946

Also related is "Proposal: 'Math for beginners' subforum" at http://mersenneforum.org/showthread.php?t=3843

THX for the first link. I shall enjoy it at my leisure.
Meanwhile, you might care to glance at a recent exchange
between Minigeek and me (Predict M45 in the lounge)
viz a viz Dr Silverman and his sensitive diplomatic posts:)

If I had to criticize your posting, I would say you didn't
appreciate leg-pulls, allowed no licence for minor misunderstandings,
didn't walk even a mile towards understanding the other person's
point of view and expounded on it at intolerable length.

David

[olivier_latinne]

Originally Posted by m_f_h

Olivier: could you show how your restrictions could be useful, e.g. to speed up the search for Mersenne primes ?
For example, to "re-discover" the known Mersenne primes ; more precisely:
For which primes p, for which mprime would start (with normal parameter setting) an LL-test, could you have foreseen that it is composite ?

Or for some primes p which are currently under investigation, could you say "easily" (even if it takes several hours or days) that Mp is not prime ?

Or something else of that kind.

m_f_h: For now, I don't think that these conjectures are actually useful to speed up the search of factors and test the primality of Mersenne numbers. Searching which form has the divisor take more time than simply apply the fast exponentiation algorithm.
But I'm cautious (not like Dr. Silverman, who jumped strait in the bobby trap), because historically there are plenty of new discoveries that we thought there were only interesting for a theoretical point of view and after a more or less long time we find that there are also very useful for a practical point of view.
But above all, I think that it is fundamental to fully understand exactly the mathematical process leading to these conjectures, and find precise demonstrations. Then, maybe, it will be very natural and easy to build more powerful tools than we have actually and explore for instance the square free conjecture or the primality of C5=M(M(7)).
It will be also important to made a generalization of the three conjectures to unspecified p (and in particularly for Fermat numbers), for composite divisors ...
Unfortunately there is almost four weeks that I have made my introduction post and no one has yet found a demonstration (Fivemack has maybe an idea but I don't know if he has made some progress? see thread #12).

There is no need to go up to 10^11, only about p=40 000 000.
But since you said you have tested the conjecture for all p<10^11, didn't give this to you new results about Mersenne primes beyond what is known?

I have been up to 10^11 for p (for j=3 and 4, M- and M+) because I think it is important to test as far as possible numerically the conjectures. I have stopped for these values of p because, with the informatics capabilities that I have, it was difficult to go above.

Anyway, you should point out what is NEW in your statement (I mean in the statement of your conjecture) : Just ONE case for which your conjecture says something more in addition to what
classical results give (in particular, e.g., the criterion explicitely given by R.D.S. in his first reply, using cubic reciprocity).
You always say "it was thought one cannot go beyond..." so you should point out WHERE *you* go beyond!

I think Fivemack (see thread #12) has already answered at the question: for some specific form (conjecture #2) of the divisors, cubic residues are always hit. The question of course is why?
All my work is based on the fact that the (non-) divisibility of a Cunningham number by a candidate divisor is deterministically linked by the form of this number.
Conjecture #1 is deterministic for all values of j. Conjecture #2 and #3 are respectively deterministic for j=3 and j=4. The special case j=12 is also deterministic by a combination of conjecture #2 and #3

PS (but this is less important): rather than posting tons of numbers in PDF files, I'd suggest to put the code (or algorithm, if you don't use PARI or Maple or so where it should be not much more than 2-3 lines) used to produce these (this is more meaningful and easier to understand) ; and point out some significant examples and counter-examples.

I have made all my programs in Fortran. All of them are a reasonable compromise between efficiency and complexity. I will give them on simple personal request.

[cheesehead]

David,

You pull your way; I'll pull mine -- perhaps we can achieve perpetual motion. :-)

[davieddy]

Originally Posted by cheesehead

David,

You pull your way; I'll pull mine -- perhaps we can achieve perpetual motion. :-)

Thanks for meeting me half way with a witty one-liner.
I feared a prolonged diatribe.
Hope this isn't tempting fate

David

[flava]

Originally Posted by olivier_latinne

There is a general principle well above physics, chemistry, biology … and mathematics too: theory need to take into account observations

Wrong, mathematics is an exact science (as opposed to physics for instance).
In physics, chemistry etc. we try to come up with a theory that covers most of the observable phenomena... theory which is often prooved wrong when our observation techniques improve. See gravity for instance:
- Newton came up with a theory based on observation. That theory is false.
- Einstein came up with a better theory which stood up quite nicely, but now requires the invention of exotic stuff like "dark matter" and "dark force" to match the observations.
- fill in here the next theory

Mathematics is never based on observation, if it is it's not mathematics. Yes, you can guess something but until a proof is found, it has no value.