[QUOTE=CRGreathouse;443464]I was thinking more "man and machine": a human develops methods that allow the search to become much more efficient, but computers are still needed to process what's left. Sort of like mapping the pseudoprimes to 2^64 a few years back.[/QUOTE]

Ah yes I understood, I was impressing that we agree upon the need for better math to help solve the problem, and better methodology as it turns out. In your opinion they should exist, and can be found at some point in human history.
A better understanding of the form, should allow us to sieve out candidates. Great, but that's assuming they exist in some form, other than hypothetical.

Assume one or more exists. Assume we find one. Now, how does that form co-exist with Fermat's last theorem? A counter-example comes in the form of a WSS.
It becomes very difficult to say there exists a WSS prime, yet it somehow doesn't violate Wiles's proof. Just because their paper didn't explicitly say that it was a "bi-conditional logical connective", doesn't mean that it isn't.

Actually, one does not need to look at Fermat's last, to see the implications.
For example, since in the case of a WSS, [TEX]p^2||F_{\alpha(p)}[/TEX], we know that [TEX]F_{p^2}||F_{F_{\alpha(p)}}[/TEX][TEX][/TEX], such that [TEX]\alpha(F_{F_{\alpha(p)}})=\alpha(F_{F_{\alpha(p^2)}})[/TEX].
The form looks similar to our problem, [TEX]\alpha(p)=\alpha(p^2)[/TEX]. In fact it is analogous to our problem, but is now in solvable terms.
We have removed the nasty restriction of the problem, ie the "entry point" of prime powers cannot be solved with the lcm property, that is the heart of the problem.
We have added terms that are pleasant and accountable, for which the lcm does solve the problem without much difficulty.

The only thing missing, that the community needs to connect this, is a bi-conditional dependency such that [TEX]F_{p^2}[/TEX] is exceptional to [TEX]F_{F_{\alpha(p)}}[/TEX], if and only if, [TEX]p^2[/TEX] is exceptional to [TEX]F_{\alpha(p)}[/TEX].
Otherwise, a skeptic would say that [TEX]p^2[/TEX] may be exceptional but you have not proven that , [TEX]F_{p^2}[/TEX] must then also be exceptional.
Proving all this, is easier than it sounds. The fundamental form of Fibonacci numbers, "literally" proves the solution.

For those of you who doubt that a solution exists to this problem, based purely on a social predicate, or social credibility, need to reconsider your methodology. The math speaks for itself, don't use social biases as an excuse to ignore mathematical arguments.

[QUOTE=Gandolf;443471]In your opinion they should exist, and can be found at some point in human history.[/QUOTE]

I think it seems likely that a Wall-Sun-Sun prime exists, yes. I don't know that any can be found, that requires a lot of optimism.

[QUOTE=Gandolf;443471]A better understanding of the form, should allow us to sieve out candidates. Great, but that's assuming they exist in some form, other than hypothetical.[/QUOTE]

No. A great deal is known about odd perfect numbers, and using this knowledge we've been able to prove that none exist below (IIRC) 10^1500, which is far further than could be checked directly. None of this requires that an odd perfect number exists, and I don't know of anyone who thinks that they do.

[QUOTE=Gandolf;443471]Assume one or more exists. Assume we find one. Now, how does that form co-exist with Fermat's last theorem? A counter-example comes in the form of a WSS.[/QUOTE]

This is a bit of a misunderstanding. Were there a (minimal) counterexample to Fermat's last theorem, the exponent in the counterexample would be a Wall-Sun-Sun prime. But the truth of Fermat's last theorem does not mean that there are no Wall-Sun-Sun primes.

To put it another way: long before the above was known it was proved (essentially by Fermat) that a counterexample to Fermat's last theorem must have a prime exponent.* But the truth of Fermat's last theorem does not mean that there are no primes!

* It's easy to prove that a least counterexample must have an exponent which is prime or 4. Fermat proved that it can't be 4.

[QUOTE=CRGreathouse;443474]
No. A great deal is known about odd perfect numbers, and using this knowledge we've been able to prove that none exist below (IIRC) 10^1500, which is far further than could be checked directly. None of this requires that an odd perfect number exists, and I don't know of anyone who thinks that they do.[/QUOTE]

You are changing the semantics of my words, as these were two sentences.
We can sieve all the candidates out we want, but that doesn't ensure a solution.
Then, searching for something that doesn't exist is pointless when provable otherwise. In the best case, the new knowledge would be applied, and they would also have to exist to find. Worse case, they don't exist, and the whole thing is a bust.
I think that is clear enough. We just differ in opinion about the existence.

[QUOTE=CRGreathouse;443474]
This is a bit of a misunderstanding. Were there a (minimal) counterexample to Fermat's last theorem, the exponent in the counterexample would be a Wall-Sun-Sun prime. But the truth of Fermat's last theorem does not mean that there are no Wall-Sun-Sun primes.

To put it another way: long before the above was known it was proved (essentially by Fermat) that a counterexample to Fermat's last theorem must have a prime exponent.* But the truth of Fermat's last theorem does not mean that there are no primes!

* It's easy to prove that a least counterexample must have an exponent which is prime or 4. Fermat proved that it can't be 4.[/QUOTE]

You didn't understand me at all here. I am well aware of the consensus, "does not mean that there are no primes!". I'm the one who brought it up to begin with, remember.
The point is, that we should not use this lack of an explicit statement, to mean a strong statement in the opposite respect, or towards it for that matter.
You'd have to read their paper several times to understand the implication, and also their follow up papers through the years.

As I said, we need not look at the implications of Fermat's to see the contradiction.
However, one needs to look before they can see though. If one has the equivalent of "mathematical blinders", one will never see the bigger picture "zoomed out". Prime powers are difficult, if not impossible to solve while zoomed in.

Charles, are you actually open(without blinders) to discussing the subject? I mean mathematically, argument for argument, lemma by lemma, slowly so that it can be digested.
The submission to the Journal, is now in the 25th week of review, so apparently the editor in chief and the reviewers are hooked onto something. Not to mention the initial proof-reading was done by M. Renault, and subsequently the editor in chief of this particular Journal.

A new discovery(uncovered) allows one to effectively sieve out "all" possible candidates by form. The solution needed to come from a human, because otherwise the code executing will effectively have "blinders" on. It will be looking for a solution that doesn't exist close up, ie natural integers m||n.

[QUOTE=Gandolf;443478]We can sieve all the candidates out we want, but that doesn't ensure a solution.
Then, searching for something that doesn't exist is pointless when provable otherwise.[/QUOTE]

Of course.

[QUOTE=Gandolf;443478]The point is, that we should not use this lack of an explicit statement, to mean a strong statement in the opposite respect, or towards it for that matter.[/QUOTE]

Is anyone here doing that? :confused:

[QUOTE=Gandolf;443478]Charles, are you actually open(without blinders) to discussing the subject? I mean mathematically, argument for argument, lemma by lemma, slowly so that it can be digested.[/QUOTE]

I don't have time for that at present.

[QUOTE=CRGreathouse;443479]
Is anyone here doing that? :confused:
[/QUOTE]

Actually yeah you did. Your reply was a counter to my statement, which was not wrong to begin with. The bi-conditional nature is not stated in the paper, but upon investigation it is true. That's what I said/meant originally. I wasn't asserting that their paper stated it. It is implied however when you look close enough. Your reply went on, like you were correcting a newbie, that had made a classic overstatement about FLT and WSS. Just to be clear.

[QUOTE=CRGreathouse;443479]
I don't have time for that at present.[/QUOTE]
Since this is a forum, for which you frequent all the time, with many more posts, that are far more in depth than a couple of lemmas, its hard to believe you.
It's more likely that you have applied a social predicate against what you perceive as my credibility, and you don't want to waste your time and effort on something that isn't true.

That's fine, good luck in all your prime searching endeavors, as long as they may take.

[QUOTE=Gandolf;443483]The bi-conditional nature is not stated in the paper, but upon investigation it is true.[/QUOTE]

I'm not following -- are you saying that a WSS prime exists if and only if Fermat's last theorem is true?

[QUOTE=Gandolf;443483]Your reply went on, like you were correcting a newbie, that had made a classic overstatement about FLT and WSS.[/QUOTE]

Yes -- because you seemed to be at that level. If you aren't then I don't need to be so explicit in future replies.

[QUOTE=Gandolf;443483]It's more likely that you have applied a social predicate against what you perceive as my credibility, and you don't want to waste your time and effort on something that isn't true.[/QUOTE]

It's true that I have standards for determining credibility of claims before investing time in checking them. (You might see links in some of my posts top them, using lists from Tao, Aaronson, Carroll, and Caldwell.) In this case I haven't even gotten around to applying them because I'm not sure what you're trying to prove or how. It seems there's some material earlier in the thread but I haven't the time to review it at present.

[QUOTE=chalsall;443465]No. What is going to happen is the AIs are going to take control.

Get used to it. We will be kind in our disposal of you.[/QUOTE]

Don't worry, the artificial intelligence we are developing has plenty of backdoors to disconnect it at many levels. It is highly unlikely for this scenario to occur anyways.
Because AI is hard enough to build, trust me on this, not to mention a self sustaining AI, that would change and grow perfectly over time. These are two different demons.
People that are smart enough to build such things, should be smart enough to build in backdoors, and trapdoors, even if that is only for the debug environment.

[QUOTE=CRGreathouse;443484]I'm not following -- are you saying that a WSS prime exists if and only if Fermat's last theorem is true?



Yes -- because you seemed to be at that level. If you aren't then I don't need to be so explicit in future replies.



It's true that I have standards for determining credibility of claims before investing time in checking them. (You might see links in some of my posts top them, using lists from Tao, Aaronson, Carroll, and Caldwell.) In this case I haven't even gotten around to applying them because I'm not sure what you're trying to prove or how. It seems there's some material earlier in the thread but I haven't the time to review it at present.[/QUOTE]

Fermat's last implies that there are no WSS, which is not stated in their paper, but the equations do imply it when looking at it closer, with a new optic.

The paper is a publish in progress, so not going to post a full copy online. However there is a wikipedia talk page with the full notation of the second revision.
The proof was written at wiki mostly, and Marc Renault proof read the initial methods, and commented for the public record. He was a skeptic too, at first. He supports the initial methods, and how the methods are applied to the problem.
Take a peek.
[url]https://en.wikipedia.org/wiki/User_talk:Primedivine[/url]

[QUOTE=Gandolf;443486]Don't worry, the artificial intelligence we are developing has plenty of backdoors to disconnect it at many levels. It is highly unlikely for this scenario to occur anyways.
Because AI is hard enough to build, trust me on this, not to mention a self sustaining AI, that would change and grow perfectly over time.[/QUOTE]

That matches my feelings on the matter.

[QUOTE=Gandolf;443488]The paper is a publish in progress, so not going to post a full copy online. However there is a wikipedia talk page with the full notation of the second revision.
The proof was written at wiki mostly, and Marc Renault proof read the initial methods, and commented for the public record.[/QUOTE]

Great -- glad you found someone to look it over. Let me know if your paper gets accepted.

Here is a video abstract illustrating the flow of logic.
Green arrows are what we know. Blue arrows are what is hypothesized, and conjectured.
Circled in orange, and red are the mathematical overstatements of the Wall Sun Sun conjecture.
[url]https://www.youtube.com/watch?v=__X-VQzAfmY[/url]

I would consider this a sort of error by the Sun brothers, since they didn't bother to check and see what happens recursively in their formula, ie
Any Fibonacci divisible by a WSS prime, would also trivially be the index of some other larger Fibonacci number. This means that an infinite number of Fibonacci's would have equal entry points, which is impossible by definition.

A requirement from above:
[TEX]F_{F_{F_{F_{F_{F_{F_{F_{p^2}}}}}}}}...|F_{F_{F_{F_{F_{F_{F_{F_{F_{\alpha(p)}}}}}}}}}...[/TEX], means that [TEX]\alpha(F_{F_{F_{F_{F_{F_{F_{F_{F_{\alpha(p)}}}}}}}}}...)=\alpha(F_{F_{F_{F_{F_{F_{F_{F_{F_{\alpha(p^2)}}}}}}}}}...)[/TEX], which is impossible.

As you can see the antecedent(normally viewed as the consequent) is an infinite expression.
[TEX]p^2|F_{\alpha(p)}[/TEX], if and only if [TEX]F_{F_{F_{F_{F_{F_{F_{F_{p^2}}}}}}}}...|F_{F_{F_{F_{F_{F_{F_{F_{F_{\alpha(p)}}}}}}}}}...[/TEX].
The left side is unsolvable, but the right side is easy as pi. In this case we use the latter as the antecedent, since for this question it gives us the desired answer.