Wall-Sun-Sun primes
 

I am curious to know the actual consensus about the existence of these primes?

Do they "probably exist", or might they exist probabilistically provided that the assumption is true, ie F(p-(p|5))/p behaving randomly modulo p?

If I understand Chris Caldwell's [URL="http://http://primes.utm.edu/glossary/xpage/WallSunSunPrime.html"]comment [/URL]correctly then the statement should depend on the assumption. I just want to clear up any ambiguity.

Does anyone know of a formula to calculate the entry point (first occurrence) of a composite factor in the Fibonacci sequence?

What is Mr. Silverman's position on the subject?

I was reading Jiri Klaska's paper, which seems to suggest a heuristic that is half of what is conjectured. Is that correct?
I'm not sure if that means that WSS primes still makes sense after klaska's adjustment.

[QUOTE=Gandolf;423331]What is Mr. Silverman's position on the subject?[/QUOTE]

this [URL="http://mersenneforum.org/member.php?u=1442"]Mr Silverman[/URL] is on a ban for right now at very least.

[QUOTE=science_man_88;423357]this [URL="http://mersenneforum.org/member.php?u=1442"]Mr Silverman[/URL] is on a ban for right now at very least.[/QUOTE]

Ok, thank you.

[QUOTE=Gandolf;422651]I am curious to know the actual consensus about the existence of these primes?[/QUOTE]

On my page
[url]https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes[/url]
I have three references on Wall-Sun-Sun primes. All agree that there should be infinitely many and that up to x you expect some multiple of log(log(x)) for large enough x. They disagree on what the multiple should be: Klaška suggests it should be 1/2, while Grell & Pend argue (more persuasively, IMO) that it should be 1. I haven't heard anyone suggest that there should be finitely many.

Thanks Charles, that makes it clearer now.

Do you have any idea why the conjecture appears named at [URL="https://oeis.org/wiki/List_of_prime_conjectures"]OEIS[/URL] as "the non-existence of Wall Sun Sun primes"?

[QUOTE=Gandolf;423427]Do you have any idea why the conjecture appears named at [URL="https://oeis.org/wiki/List_of_prime_conjectures"]OEIS[/URL] as "the non-existence of Wall Sun Sun primes"?[/QUOTE]

I don't, but I can see in the revision history that the particular entry was written by a high-school student so I would take it [i]cum grano salis[/i]. That particular page hasn't been reviewed yet -- most of the OEIS editors focus on the sequences rather than the wiki.

[QUOTE=CRGreathouse;423467]I don't, but I can see in the revision history that the particular entry was written by a high-school student so I would take it [I]cum grano salis[/I]. That particular page hasn't been reviewed yet -- most of the OEIS editors focus on the sequences rather than the wiki.[/QUOTE]

Alright, the note section may indicate the reason why he named it that. Could you please read the references within it, and verify, since the name is changed now, it does not look consistent with the old reference.

[URL]http://arxiv.org/pdf/1102.1636v2.pdf[/URL]

"The Wall-Sun-Sun prime conjecture is as follows,..There does [B]not[/B] exist a prime p such that p^2 | F(p-(p|5))".

[QUOTE=Gandolf;423472]Alright, the note section may indicate the reason why he named it that. Could you please read the references within it, and verify, since the name is changed now, it does not look consistent with the old reference.

[URL]http://arxiv.org/pdf/1102.1636v2.pdf[/URL]

"The Wall-Sun-Sun prime conjecture is as follows,..There does [B]not[/B] exist a prime p such that p^2 | F(p-(p|5))".[/QUOTE]

The paper you cite was written, apparently, by two undergrads (sophomores, the paper says). It doesn't source the conjecture.

The Sun-Sun paper
[url]http://matwbn.icm.edu.pl/ksiazki/aa/aa60/aa6046.pdf[/url]
doesn't make this conjecture. The Williams paper
[url]http://www.sciencedirect.com/science/article/pii/0898122182900268[/url]
says that "Wall's problem is to find a p such that ...", and suggests the 1/p heuristic which suggests infinitely many exist.

Peng
[url]http://arxiv.org/abs/1511.05645[/url]
though says that Wall conjectured (something equivalent to the nonexistence of these primes). I don't have a copy of Wall's paper at the moment, but if so then this should properly be called Wall's conjecture rather than W-S-S since the latter two do not join him.

[QUOTE=science_man_88;423357]this [URL="http://mersenneforum.org/member.php?u=1442"]Mr Silverman[/URL] is on a ban for right now at very least.[/QUOTE]
Discussion on this moved to [url="http://mersenneforum.org/showthread.php?t=20895"]here[/url]

[QUOTE=jasonp;423636]Discussion on this moved to [URL="http://mersenneforum.org/showthread.php?t=20895"]here[/URL][/QUOTE]
Thanks.

[QUOTE=CRGreathouse;423604]The paper you cite was written, apparently, by two undergrads (sophomores, the paper says). It doesn't source the conjecture.

The Sun-Sun paper
[URL]http://matwbn.icm.edu.pl/ksiazki/aa/aa60/aa6046.pdf[/URL]
doesn't make this conjecture. The Williams paper
[URL]http://www.sciencedirect.com/science/article/pii/0898122182900268[/URL]
says that "Wall's problem is to find a p such that ...", and suggests the 1/p heuristic which suggests infinitely many exist.

Peng
[URL]http://arxiv.org/abs/1511.05645[/URL]
though says that Wall conjectured (something equivalent to the nonexistence of these primes). I don't have a copy of Wall's paper at the moment, but if so then this should properly be called Wall's conjecture rather than W-S-S since the latter two do not join him.[/QUOTE]

Ah, thank you very much. I knew something wasn't consistent across the papers, because it was hard to relate the question to the conjecture.

Wall asked if u(p)=u(p^2) is always impossible, from what I've read. Semantically, it could imply existence, but I would assume the question would be this instead: Is u(p)=u(p^2) ever possible?

I do think Wall's conjecture was meant to imply a pattern of non-existence, yet open to the possibility of existence. Apparently, Wall's question is equivalent to, Is [TEX]F_{(p^2)}\mid F_{(F_{(u_{p^1})})} [/TEX] always impossible?


If an integer [TEX]m[/TEX], has prime factorization, [TEX]p_{1}^{e_{1}}\cdot\ p_{2}^{e_{2}}...p_{n}^{e_{n}}[/TEX] then the "entry point"(index) of [TEX]m[/TEX] equals, [TEX]u_{m} = \operatorname{lcm}({ u_{p_{1}^{e_{1}}}, u_{p_{2}^{e_{2}}}, ... u_{p_{n}^{e_{n}}}}) [/TEX].

Consider [TEX]m=F_{(F_{(u_{p^1})})} [/TEX], where the prime factorization is grouped into packets of [TEX]F_{(p^e)}[/TEX], except for any product of primitive factors, which always have an entry point of [TEX]F_{(u_{p^1})}[/TEX].

Suppose [TEX]p_{1}[/TEX] is a Fibonacci Wieferich prime, then [TEX]m=(F_{(p_{1}^2)}\ \cdot\ F_{(p_{2}^{e_{2}})}\ \cdot\ ...\ F_{(p_{n}^{e_{n}})}\ \cdot\ \prod{})=F_{(F_{(u_{p^1})})}[/TEX].

The entry point of [TEX]m=F_{(F_{(u_{p^1})})} [/TEX] is always supposed to be [TEX]F_{(u_{p^1})} [/TEX] never [TEX]F_{(p*u_{p^1})} [/TEX].

However in this case, [TEX]u_m=\operatorname{lcm}(p_{1}^{2},\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}},\ F_{(u_{p^1})})\ \neq F_{(u_{p^1})}=F_{(p*u_{p^1})}[/TEX].

[TEX]F_{(p^2)}[/TEX] cannot divide [TEX]F_{(F_{(u_{p^1})})}[/TEX], together with those other factors.

[QUOTE=Gandolf;423708]Wall asked if u(p)=u(p^2) is always impossible, from what I've read. Semantically, it could imply existence, but I would assume the question would be this instead: Is u(p)=u(p^2) ever possible?

I do think Wall's conjecture was meant to imply a pattern of non-existence, yet open to the possibility of existence. Apparently, Wall's question is equivalent to, Is [TEX]F_{(p^2)}\mid F_{(F_{(u_{p^1})})} [/TEX] always impossible?[/QUOTE]

I'll have to get a copy of the paper to check if he posed it as an open question or if he made a conjecture.

[QUOTE=CRGreathouse;423787]I'll have to get a copy of the paper to check if he posed it as an open question or if he made a conjecture.[/QUOTE]

Page 528 I think.
[QUOTE]Remark. The most perplexing problem we have met in this study concerns the hypothesis [TEX]k_{p^2}\text{ not equal to } k_{p}[/TEX]. We have run.... ....however cannot yet prove that [TEX]k_{p^2} = k_{p}[/TEX] is impossible. The question is closely related to... can a number x have the same order mod p and mod p^2?...; hence one might conjecture that equality may hold for some exceptional p.[/QUOTE]

I wasn't able to obtain copyright permission to post the details here, but if anyone is interested in reading his original paper, free see here: [URL]http://www.jstor.org/stable/2309169?seq=1#page_scan_tab_contents[/URL]

The paper starts:

[QUOTE]This [B]inquiry[/B] is concerned with determining the length of the period... The [B]problem[/B] arose in connection with a method for generating random numbers, but it turned out to be unexpectedly intricate, and so quickly became of interest in its own right....
At least two [B]questions[/B] remain unanswered... Remark 1, and 2.

Remark 2.
Theorems 6 and 7 furnish upper bounds for the function k(p), and we easily find cases where k(p) has these maximum values. On the other hand, we cannot find a nontrivial lower bound for k(p)...
[/QUOTE]
If you read theorem 2, it shows the entry point relationship, which is stated more clearly(conveniently) in M. Renault's 1996 paper. [URL]http://webspace.ship.edu/msrenault/fibonacci/FibThesis.pdf[/URL] [URL]http://webspace.ship.edu/msrenault/fibonacci/fib.htm[/URL]
Marc has not found any flaws in his theorem, nor in this application of it, relating to the proof. Although he is still looking at it closely.

Proof
 

Let [TEX]F_{(u_{p^1})}[/TEX], denote the least positive Fibonacci number divisible by the prime [TEX]p[/TEX].
Let [TEX]F_{(u_{p^2})}[/TEX], denote the least positive Fibonacci number divisible by [TEX]p^2[/TEX].
Let [TEX]\prod_{1}^{n\ge 1}[/TEX] be a quotient of primitive(characteristic) prime factor(s), ie factors that have not occurred in any earlier Fibonacci numbers.
This may be represented below as an empty product of one, if the quotient divides [TEX]F_{(p^{e}_{1})},\ e>1[/TEX].

[TEX]{n \mid m}\ \text{iff}\ {F_{(n)}\mid F_{(m)}}\, n \ge 3 [/TEX]

[TEX]p^2 \mid F_{(u_{p^1})}\ \text{iff}\ F_{(p^2)} \mid F_{(F_{(u_{p^1})})}[/TEX]
If an integer [TEX]i[/TEX], has prime factorization, [TEX]p_{1}^{e_{1}}\cdot\ p_{2}^{e_{2}}...p_{n}^{e_{n}}[/TEX] then the entry point of [TEX]i[/TEX] equals, [TEX]u_{i} = \operatorname{lcm}[{ u_{(p_{1}^{e_{1}})}, u_{(p_{2}^{e_{2}})}, ...u_{(p_{n}^{e_{n}})}}][/TEX].

[TEX]\text{If}\ i=F_{(F_{(u_{p^1})})}\ \text{then, }\ F_{(p^2)} \not| F_{(F_{(u_{p^1})})}\text{ because } u_{F_{(F_{(u_{p^1})})}}\neq u_{F_{(F_{(u_{p^2})})}} [/TEX].

An abstract example of a Fibonacci-Wieferich prime p, where [TEX]p_{1}=p,\ e_{1}\ge 2[/TEX].
If [TEX]i=F_{(F_{(u_{p^1})})}=(F_{(p_{1}^2)}\ \cdot\ F_{(p_{2}^{e_{2}})}\ \cdot\ ...\ F_{(p_{n}^{e_{n}})}\ \cdot\ \prod{})[/TEX] then, can the entry point of [TEX]i[/TEX] still equal the same index, ie [TEX]u_{(F_{(F_{(u_{p^1})}}))}=F_{(u_{p^1})}[/TEX]?
Answer: No.
Proof: ''This is based upon the observation that the entry point of the product of primitive prime factors, is supposed to be equal to the entry point of the product of non-primitive prime factors, ie Fibonacci numbers with unique factorization for indices, in this case.''

The factor, [TEX]j=(F_{(p_{1}^1)}\ \cdot\ F_{(p_{2}^{e_{2}})}\ \cdot\ ...\ F_{(p_{n}^{e_{n}})}\ \cdot\ \prod{})[/TEX] always has an entry point of, [TEX]u_j=\operatorname{lcm}[p_{1}^1,\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}},\ F_{(u_{p^1})}]=\operatorname{lcm}[p_{1}^1,\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}}]=F_{(u_{p^1})}[/TEX].
While the factor [TEX]i[/TEX] has a later entry point, [TEX]u_i=\operatorname{lcm}[p_{1}^{2},\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}},\ F_{(u_{p^1})}]=\operatorname{lcm}[p_{1}^{2},\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}}]\neq F_{(u_{p^1})}=F_{(p\cdot u_{p^1})}[/TEX]

[TEX]F_{(u_{p^1})}\neq F_{(u_{p^2})}[/TEX]

[TEX]u_{(p^1)}\neq u_{(p^2)}[/TEX]

[TEX]u_{(p^2)} = p\cdot u_{(p^1)}[/TEX]

[TEX]p^{2}\not| F_{(u_{p^1})}[/TEX]

Charles, what is your opinion on the subject now?

[QUOTE=CRGreathouse;423787]I'll have to get a copy of the paper to check if he posed it as an open question or if he made a conjecture.[/QUOTE]

Hey Charles, have you gotten a copy of the paper yet? What do you think, is it an open question or, did he make a conjecture in your opinion?

Do you think a counter-example, aka Wall-Sun-Sun prime will eventually be found with brute force?

[QUOTE=Gandolf;443391]Hey Charles, have you gotten a copy of the paper yet? What do you think, is it an open question or, did he make a conjecture in your opinion?[/QUOTE]

I don't think I ever found a copy. I'm not comfortable guessing what he may have had in mind.

[QUOTE=Gandolf;443391]Do you think a counter-example, aka Wall-Sun-Sun prime will eventually be found with brute force?[/QUOTE]

It's not terribly likely. PrimeGrid has searched for one using lots of computer power without finding one. If by a massive effort they searched a thousand times as far, the heuristic chances they'd find one is around 8% if I recall properly. It's not terrible but not great either, and that would be a huge effort. It doesn't look like Moore's law can help us either. More math might, though -- I could imagine more properties being found which allowed a search (possibly non-exhaustive) much further.

[QUOTE=CRGreathouse;443418]
It's not terribly likely. PrimeGrid has searched for one using lots of computer power without finding one. If by a massive effort they searched a thousand times as far, the heuristic chances they'd find one is around 8% if I recall properly. It's not terrible but not great either, and that would be a huge effort. It doesn't look like Moore's law can help us either. More math might, though -- I could imagine more properties being found which allowed a search (possibly non-exhaustive) much further.[/QUOTE]

True, and Primegrid had the WSS bug. If I am not mistaken this invalidates a lot of the results, and they'd need to be re-checked. The exact heuristics are in question, from what I've read. Things have changed over the years, although one of the Sun brothers "apparently" contacted a Primegrid member, after that last false positive, and was confident that we would find a WSS soon. Honestly, I have not heard the same answer from any two professors. Seriously.

I agree that better math is needed. This problem is almost like "man versus machine".
A human seems the most likely to spit out the correct answer.

[QUOTE=Gandolf;443455]I agree that better math is needed. This problem is almost like "man versus machine".
A human seems the most likely to spit out the correct answer.[/QUOTE]

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=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]

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=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.

[QUOTE=Gandolf;443672]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.[/QUOTE]

It's this kind of slippery logic that has me worried for the safety of your proof.

Explain your comment then. What have you misunderstood about the quote?

I never use slippery logic of any kind. In this case it is the mathematical community that has jumped to conclusions about WSS. I did para-phrase the solution for you, to make it easier to understand. My reward was to be thrown under the bus with my alleged slippery logic.

Charles, I hope it's okay for me to say this without you getting mad, but you are nowhere near as smart as you think you are.

I did have a chance to review Wall's paper today. I would say that he conjectured (weakly) that Wall-Sun-Sun primes exist, though he didn't venture to say if he thought there were infinitely many.

I haven't yet seen anything from you that suggests that you have a proof that there are no Wall-Sun-Sun primes. Since you have already have someone who has looked over your proof and have already submitted the result to a journal (which one?), I don't see a need to look it over further until it's published. Hopefully that version will be clearer.

[QUOTE=CRGreathouse;443720]I haven't yet seen anything from you that suggests that you have a proof that there are no Wall-Sun-Sun primes.
...Hopefully that version will be clearer.[/QUOTE]

Seeing, as one must look before they can see, I understand why you haven't seen anything.
You final comment was more of a passive aggressive one, since you admit you haven't looked, yet you imply that it isn't clear enough. Just speak the truth about what you know. I certainly don't need your slippery approval, and wild guesses.
Unless you have anything constructive to say, we are done.

[QUOTE=Gandolf;443721]Just speak the truth about what you know.[/QUOTE]

What I have seen of your proof does not inspire confidence in the correctness of your claims. I have spent about 20 minutes looking through your proof between your posts here and the Wikipedia page. The last manuscript I reviewed for a journal took me about 15 hours to referee. I'm happy keeping my time investment closer to the former than the latter here.

[QUOTE=CRGreathouse;443735]What I have seen of your proof does not inspire confidence in the correctness of your claims. I have spent about 20 minutes looking through your proof between your posts here and the Wikipedia page. The last manuscript I reviewed for a journal took me about 15 hours to referee. I'm happy keeping my time investment closer to the former than the latter here.[/QUOTE]

The hubris of the defeated. Spends, 20 minutes, finds absolutely no objections to any of the mathematical arguments. Then claims that the proof is somehow at fault, for his lack of something to object to. lol Classic

Gandolf, you are coming across as a surly crank.

[QUOTE=GP2;443743]Gandolf, you are coming across as a surly crank.[/QUOTE]
Sure, that was Charles's goal.

Could you keep on discussing math, and not resort to personal attacks?

Now let me ask a stupid question. I can not exactly visualize how this WSS primes look like. Are WSS primes also Wieferich primes? Does it mean that if we find a WSS prime, we'll also have a third Wieferich prime?

[QUOTE=LaurV;443753]Could you keep on discussing math, and not resort to personal attacks?

Now let me ask a stupid question. I can not exactly visualize how this WSS primes look like. Are WSS primes also Wieferich primes? Does it mean that if we find a WSS prime, we'll also have a third Wieferich prime?[/QUOTE]

I absolutely can talk to anyone that is polite and objective. I certainly didn't attack Charles first.

That's a good question as simple as it is.
The heuristic is based on that probability.
[QUOTE]Page 528
Wall Quote:
Remark. The most perplexing problem we have met in this study concerns the [B]hypothesis[/B] k_{p^2}\text{ not equal to } k_{p}.
We have run a test on digital computer.... ....however cannot [B]yet[/B] prove that k_{p^2} = k_{p} is impossible.
The question is closely related to... can a number x have the same order mod p and mod p^2?..., for which rare cases give an affirmative answer(e.g., x=3, p=11; x=2, p=1093); hence [I]one might conjecture[/I] that equality [I]may[/I] hold for some [B]exceptional[/B] p. [/QUOTE]

This is the origin of the split between his strong hypothesis, and the weaker conjecture that one(someone else;Sun Sun) might assert.
Notice he does not officially assert the conjecture himself. The paper shows nothing about this weaker conjecture, and is focused entirely on the hypothesis at hand. Although he admits that the question is open, since he could not prove otherwise.

Just figured I'd re-iterate that it was an open question, with a strong hypothesis, not just a weak conjecture as stated by Charles.

[QUOTE=Gandolf;443719]Charles, I hope it's okay for me to say this without you getting mad, but you are nowhere near as smart as you think you are.[/QUOTE]
[QUOTE=Gandolf;443758] I certainly didn't attack Charles first.
[/QUOTE]
You didn't?

Maybe you misunderstood the concept of the [URL="https://en.wikipedia.org/wiki/Slippery_slope"]slippery slope[/URL]?
Slippery slope has to do with the logic of the argument. It has nothing to do with "attacking you"[B].
[/B]
In contrast, these are not just one but two personal attacks:
[QUOTE=Gandolf;443719]...without you getting mad, [/QUOTE][QUOTE=Gandolf;443719] ...but you are nowhere near as smart as you think you are.[/QUOTE]
Comments?

[QUOTE=Batalov;443761]You didn't?

Maybe you misunderstood the concept of the [URL="https://en.wikipedia.org/wiki/Slippery_slope"]slippery slope[/URL]?
Slippery slope has to do with the logic of the argument. It has nothing to do with "attacking you"[B].
[/B]
In contrast, these are not just one but two personal attacks:

Comments?[/QUOTE]

No, it was preceded by open prejudice against an amateur with a solution to a difficult problem. That is offensive, and a very common form of bigotry, in the field. It's bullshit. Fuck that. I will be speaking about it after I publish. I will not be speaking kindly of him.

If you call something a slippery slope, a common phrase, you are absolutely obligated say something about what you are referring to. Slippery logic as far as I know is being stupid, silly, careless with an argument etc. The wiki link didn't apply to anything I said.
He offered no reason why he would say that. You have crazy ideas, ......what ideas are crazy? .....silence....

Take Charles's word "with a grain of salt". Take it as an insult, but don't worry, Charles is famous for saying this about others, even without reading their papers. That is when I immediately knew that he was an ass. You can't judge someone's math without reading the paper. You can't just base your judgment on some wacky social predicate.

Anyways, it has nothing to do with WSS.

[QUOTE=Gandolf;443758]Just figured I'd re-iterate that it was an open question, with a strong hypothesis, not just a weak conjecture as stated by Charles.[/QUOTE]

By "weak conjecture" I meant only that Wall wasn't expressing a high degree of certainty.

[QUOTE=Gandolf;443763]If you call something a slippery slope, a common phrase, you are absolutely obligated say something about what you are referring to. [/QUOTE]
Except if one is replying to the previous line, which he did.

[QUOTE=Gandolf;443672]...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.
[/QUOTE]
That, above, [B]is[/B] a [URL="https://en.wikipedia.org/wiki/Slippery_slope"]slippery slope argument[/URL]. The argument slips from singular to an infinite expression.

[COLOR=DarkRed]Now listen carefully - one more personal attack will lead to a permanent ban. But for now, a week ban to sit back and think twice before calling someone who was foolishly helping you - the pile of ugly things that you just dumped in the previous message.

:banned:
[/COLOR]

[QUOTE=Gandolf;443763]You can't judge someone's math without reading the paper. You can't just base your judgment on some wacky social predicate.[/QUOTE]

Sure you can -- and must, if you're to make any headway through the vast number of published and unpublished papers today. Here are three links dealing with this sort of mental heuristic, from gentlest to harshest:
[url]https://terrytao.wordpress.com/career-advice/dont-prematurely-obsess-on-a-single-big-problem-or-big-theory/[/url]
[url]http://www.scottaaronson.com/blog/?p=304[/url]
[url]http://primes.utm.edu/notes/crackpot.html[/url]

Not everyone has the courage these three show by posting these guidelines, but everyone -- by virtue of the sheer amount of material out there -- has to have some principles on what they choose to read. Maybe they prefer to read whatever they come across first, maybe they read only what has been recommended to them in person, maybe they read only what is assigned in their classes, but somehow or other they have to decide between reading [url=http://timecube.2enp.com/]this[/url], [url=https://arxiv.org/abs/math/0606088]this[/url], and everything else.

I'm sorry you're so offended by my application of my heuristics, or in explaining such frankly. I imagine many others merely passed the thread by rather than risk comment. Certainly I leave open the possibility that you have some great discovery which I have passed over in error. (There's always a Type I/Type II tradeoff.) I will revisit my decision if and when your paper is accepted for publication in a reputable journal.

As for the heuristics themselves, I would be interested to hear your thoughts on distinguishing papers which are worth your time to read from those which are not. This would belong in a new thread, naturally.

[QUOTE=Gandolf;424680]An abstract example of a Fibonacci-Wieferich prime p, where [TEX]p_{1}=p,\ e_{1}\ge 2[/TEX].
If [TEX]i=F_{(F_{(u_{p^1})})}=(F_{(p_{1}^2)}\ \cdot\ F_{(p_{2}^{e_{2}})}\ \cdot\ ...\ F_{(p_{n}^{e_{n}})}\ \cdot\ \prod{})[/TEX] then, can the entry point of [TEX]i[/TEX] still equal the same index, ie [TEX]u_{(F_{(F_{(u_{p^1})}}))}=F_{(u_{p^1})}[/TEX]?
Answer: No.
Proof: ''This is based upon the observation that the entry point of the product of primitive prime factors, is supposed to be equal to the entry point of the product of non-primitive prime factors, ie Fibonacci numbers with unique factorization for indices, in this case.''

The factor, [TEX]j=(F_{(p_{1}^1)}\ \cdot\ F_{(p_{2}^{e_{2}})}\ \cdot\ ...\ F_{(p_{n}^{e_{n}})}\ \cdot\ \prod{})[/TEX] always has an entry point of, [TEX]u_j=\operatorname{lcm}[p_{1}^1,\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}},\ F_{(u_{p^1})}]=\operatorname{lcm}[p_{1}^1,\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}}]=F_{(u_{p^1})}[/TEX].
While the factor [TEX]i[/TEX] has a later entry point, [TEX]u_i=\operatorname{lcm}[p_{1}^{2},\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}},\ F_{(u_{p^1})}]=\operatorname{lcm}[p_{1}^{2},\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}}]\neq F_{(u_{p^1})}=F_{(p\cdot u_{p^1})}[/TEX]

[/QUOTE]

This looks suspect. By hypothesis, p1^e1 divides F(u(p1)). Hence lcm[p1, p2^e2, ... , F(u(p1))] = p1^e1*p2^e2... = F(u(p1)). But lcm[p1^2, p2^e2, ... , F(u(p1))] will also be p1^e1*p2^e2 ... = F(u(p1)).

[QUOTE=axn;444202]This looks suspect. By hypothesis, p1^e1 divides F(u(p1)). Hence lcm[p1, p2^e2, ... , F(u(p1))] = p1^e1*p2^e2... = F(u(p1)). But lcm[p1^2, p2^e2, ... , F(u(p1))] will also be p1^e1*p2^e2 ... = F(u(p1)).[/QUOTE]

Yes. The paper and subsequent refinements have been reviewed by several journals although unfortunately none of them decided to publish the results based on conflicts of interest, and the controversial nature of the subject matter. The consensus is that both the main idea and stratagem appear to be correct though. The paper now splits the fundamental theorem of arithmetic into 4 groups, in order to prove the final collective result in terms of Fibonacci numbers rather than natural numbers. These kinds of questions may very well not be answerable in terms of natural numbers, directly.

Dr. Carl Pomerance has semi-officially proof-read the paper and he made some very good suggestions that made it into the revised paper. Both Marc Renault and Pomerance appear in the acknowledgment section for their contributions.

This paper remains a working draft.
[url]https://www.dropbox.com/s/bkdkhhmnexyynd6/Wall.pdf?dl=0[/url]