Talk:Wall–Sun–Sun prime

Dubious statement

I don't really get the purpose following statement:

"As a result, prior to Andrew Wiles' proof of Fermat's last theorem, the search for Wall–Sun–Sun primes was also the search for a counterexample to this centuries-old conjecture."

In which way would the discovery of a Wall-Sun-Sun prime constitute a counterexample to Fermat's last theorem (or the First Case, as I guess that's what is meant here)? -- Toshio Yamaguchi (tlk−ctb) 08:33, 22 July 2012 (UTC)

It wouldn't necessarily have been a counterexample so I have changed it to "the search for a potential counterexample". PrimeHunter (talk) 09:26, 22 July 2012 (UTC)

I think that's better. Thanks. -- Toshio Yamaguchi (tlk−ctb) 09:30, 22 July 2012 (UTC)

Why is it conjectured true?

If no such prime has been found, why are they conjectured to exist? There must be some justification, and if it would fit into the article I'd be curious to know. — Preceding unsigned comment added by 24.3.189.16 (talk) 13:24, 11 November 2012 (UTC)

I think that conjecture is based on the assumption that the remainders of $F_{p-\left({\frac {p}{5}}\right)}$ modulo p² are uniformly distributed in the set of integers {0, 1, …, p–1}. By that reasoning, the chance the the remainder is 0 is about ${\tfrac {1}{p}}$ and summation of the values of ${\tfrac {1}{p}}$ for all primes p in an interval [x, y] leads to the formula $\sum _{x\leq p\leq y}{\tfrac {1}{p}}$ which is approximately $\ln {\Big (}{\tfrac {\ln(y)}{\ln(x)}}{\Big )}$ and this grows to infinity as p goes to infinity. Therefore, if one assumes x to be fixed at say 1 and lets y grow to infinity, then this logarithmic expression also goes to infinity. So if this heuristic is correct (and I don't think it has been proven that it is), then there should exist infinitely many Wall-Sun-Sun primes, which means they would indeed exist. -- Toshio Yamaguchi (tlk−ctb) 14:09, 11 November 2012 (UTC)

Existence

Let $F_{u_{p^{1}}}$, be the smallest Fibonacci number divisible by the prime $p$.

Let $F_{u_{p^{2}}}$, be the smallest Fibonacci number divisible by $p^{2}$.

Let $\prod _{1}^{n\geq 1}$ be the quotient of primitive(characteristic) prime factor(s), ie factors that have not occurred in any earlier Fibonacci numbers.

$\left\lbrace {n\mid m}\ {\text{iff}}\ {F_{n}\mid F_{m}}\right\rbrace \,n\geq 3$

$p^{2}\mid F_{u_{p^{1}}}\ {\text{iff}}\ F_{p^{2}}\mid F_{F_{u_{p^{1}}}}$

If an integer $i$, has prime factorization, $p_{1}^{e_{1}}\cdot \ p_{2}^{e_{2}}...p_{n}^{e_{n}}$ then the entry point of $i$ equals, $\left\lbrace u_{i}=\operatorname {lcm} ({u_{p_{1}^{e_{1}}},u_{p_{2}^{e_{2}}},u_{p_{n}^{e_{n}}}})\right\rbrace$.

${\text{If}}\ i=F_{F_{u_{p^{1}}}}\ {\text{then}}\ F_{p^{2}}\nmid \!F_{F_{u_{p^{1}}}}\ \left\lbrace F_{F_{u_{p^{1}}}}\neq F_{F_{u_{p^{2}}}}\right\rbrace$.

$example:$

$p=71,u_{p}=70,F_{70}=190392490709135$

$j=(F_{71}\ \cdot \ F_{5}\ \cdot \ F_{11}\ \cdot \ F_{13}\ \cdot \ F_{29}\ \cdot \ F_{911}\ \cdot \ F_{141961}\ \cdot \ \prod {})$

$i=(F_{71^{2}}\ \cdot \ F_{5}\ \cdot \ F_{11}\ \cdot \ F_{13}\ \cdot \ F_{29}\ \cdot \ F_{911}\ \cdot \ F_{141961}\ \cdot \ \prod {})$

$u_{j}=\operatorname {lcm} (71,\ 5,\ 11,\ 13,\ 29,\ 911,\ 141961,\ 190392490709135)=190392490709135=F_{u_{p^{1}}}$

$u_{i}=\operatorname {lcm} (71^{2},\ 5,\ 11,\ 13,\ 29,\ 911,\ 141961,\ 190392490709135)=13517866840348585=F_{pu_{p^{1}}}$

${\text{Abstract example, }}p_{1}=p,\ e_{1}\geq 2$

${\text{If }}i=(F_{p_{1}^{2}}\ \cdot \ F_{p_{2}^{e_{2}}}\ \cdot \ ...\ F_{p_{n}^{e_{n}}}\ \cdot \ \prod {})=F_{F_{u_{p^{1}}}}\ {\text{then does i's entry point }}u_{(F_{F_{u_{p^{1}}}})}=F_{u_{p^{1}}}\ {\text{still?}}$

${\text{No, since any extra powers of p produce a later entry point, }}u_{i}=\operatorname {lcm} (p_{1}^{2},\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}},\ F_{u_{p^{1}}})\neq F_{u_{p^{1}}}=F_{pu_{p^{1}}}.$

${\text{While the factor, }}j=(F_{p_{1}^{1}}\ \cdot \ F_{p_{2}^{e_{2}}}\ \cdot \ ...\ F_{p_{n}^{e_{n}}}\ \cdot \ \prod {}){\text{ always has an entry point of, }}u_{j}=\operatorname {lcm} (p_{1}^{1},\ p_{2}^{e_{2}},\ ...\ p_{n}^{e_{n}},\ F_{u_{p^{1}}})=F_{u_{p^{1}}}.$

${F_{u_{p^{1}}}}\neq {F_{u_{p^{2}}}}$

${u_{p^{1}}}\neq {u_{p^{2}}}$

${u_{p^{2}}}={pu_{p^{1}}}$

$p^{2}\nmid \!F_{u_{p^{1}}}$

This shows that Zhi-Hong Sun and Zhi-Wei Sun's equation always returns false.

${\text{If}}\ p\ \mid \ xyz\ {\text{and also}}\ x^{p}+y^{p}=z^{p}\ {\text{then}}\ p\ {\text{is a Wall-Sun-Sun prime.}}$

so

${\text{If}}\ p\ \mid \ xyz\ {\text{then}}\ x^{p}+y^{p}\neq z^{p}\ {\text{since there are no Wall-Sun-Sun primes.}}$

• $\operatorname {gcd} (F_{p},F_{n})=F_{gcd(p,n)}=F_{1}=1,\ {\text{for}}\ n<p$
• $\operatorname {gcd} (F_{F_{p}},F_{F_{n}})=F_{gcd(F_{p},F_{n})}=F_{F_{1}}=1,\ {\text{for}}\ n<p$
• $\operatorname {gcd} (F_{F_{F_{p}}},F_{F_{F_{n}}})=F_{gcd(F_{F_{p}},F_{F_{n})}}=F_{F_{F_{1}}}=1,\ {\text{for}}\ n<p$
• $\operatorname {gcd} (F_{F_{F_{._{._{._{F_{p}}}}}}},F_{F_{F_{._{._{._{F_{n}}}}}}})=F_{gcd(F_{F_{._{._{._{F_{p}}}}}},F_{F_{._{._{._{F_{n}}}})}}}=F_{F_{F_{._{._{._{F_{1}}}}}}}=1,\ {\text{for}}\ n<p$

• $n\mid m$ iff $F_{n}\mid F_{m}$
• $n\mid m$ iff $F_{F_{n}}\mid F_{F_{m}}$
• $n\mid m$ iff $F_{F_{F_{n}}}\mid F_{F_{F_{m}}}$
• $n\mid m$ iff $F_{F_{._{._{._{F_{n}}}}}}\mid F_{F_{._{._{._{F_{m}}}}}}$

• $p^{2}\mid F_{u_{p^{1}}}$ iff $F_{F_{._{._{._{F_{p^{2}}}}}}}\mid F_{F_{._{._{._{F_{F_{u_{p^{1}}}}}}}}}$Fibonacci-Wieferich

An earlier discussion is archived at WP:Reference desk/Archives/Mathematics/2015 December 8#Can Wall-Sun-Sun primes exist?.

@Primedivine:

1: Do you believe that, if the logic above is correct, then it constitutes a proof that there are no Wall-Sun-Sun primes?

2: Are you planning on publishing this work elsewhere?

-- ToE 15:20, 27 December 2015 (UTC)

1: I should be careful to point out, that my belief is separate from my observation that the article lacks balance as an open question. Although, my answer is yes because the lcm is a known property for all composite integers, and it exposes an extra factor of p, in the compared entry points.

2: I don't consider this work(original research), but merely a trivial observation that has been overlooked. In the Wikipedia rules this is exempt from original research, because the calculation is trivial, using only multiplication and division. You can see why it was overlooked, because usually the logic F(n)|F(m) iff n|m applies, where the indices yield the divisibility property of the Fibonacci numbers, but the statement works to our advantage in reverse, which is counter-intuitive, thus overlooked. Some of the last few lines may be contested because it actually closes the open part of the question, rather than just implying a vicarious imposition of F(n),F(m).

An edit will have to wait for publication. Primedivine (talk) 14:39, 19 January 2016 (UTC)

Equvalents for k-Wall-Sun-Sun

I don't understand the first terms in the article under the section "Generalizations" being "equivalent". The Lucas sequence of the second type V_n(k=2,-1) is

V_0(2,-1)=2,

V_1(2,-1)=2,

V_2(2,-1)=6,

V_3(2,-1)=14,

...,

V_13(2,-1) = 94642

as registered in the article on Lucas sequences.

Now at p=13 V_p(2,-1)=94642 mod (13^2) = 2, but according to the definition this needs to be 1 if 13 is supposed to be the smallest k-Wall-Sun-Sun prime for k=2 (?). R. J. Mathar (talk) 16:56, 21 April 2016 (UTC)

• The definition of k-Wall-Sun-Sun prime is now corrected. Maxal (talk) 16:04, 22 April 2016 (UTC)

Inconsitency in Wall–Sun–Sun primes with discriminant D

Currently, the article states that

a Lucas–Wieferich prime p associated with (P, Q) is ... a Wall–Sun–Sun prime with discriminant D = P² – 4Q.

However, this is apparently is inconsistent with the primes listed in the table (copied from the Elsenhans 2010 reference for square-free D, as I understand). In particular, prime 3 was missing in the table for D = 20, while it is a Lucas–Wieferich prime associated with (4,-1). Either definition of the Wall–Sun–Sun prime with discriminant D or the table values need to be updated.

Another issue is that the current definition is said to apply only for odd pand not dividing D. It is not clear if p = 2 and/or dividing D are absolutely excluded from being Wall–Sun–Sun prime with discriminant, or if it is just a restriction of the current definition (which others may be applied for such primes). Maxal (talk) 16:52, 25 April 2016 (UTC)

Definition

Where is the reference showing that pisano periods are part of the definition of Wall Sun Sun primes? That is not how Donald Dines Wall originally defined the problem.

This is original research without a citation. The proper term is entry point, or the symbol alpha, ie $\alpha (p)$ is the subscript of the least positive Fibonacci number divisible by p.

I can see why one would conclude that though, because $\alpha (p)\mid \mid \pi (p)$. But that doesn't justify it, without a reference. Honestly even with a reference it is not the standard terminology that specifically and accurately defines the problem. Primedivine (talk) 19:20, 7 May 2016 (UTC)

• The reference (Elsenhans and Jahnel, 2010) is given in the article. Maxal (talk) 21:18, 22 May 2016 (UTC)

Unfortunately, I cannot find anything about pisano periods in that paper. This part of the article should be removed and corrected, since it is not part of the definition whatsoever. Original research is not allowed.

No experts have claimed that pisano periods define Wall Sun Sun primes. It is something that happens as a consequent(if they exist), but it is irrelevant to the definition. Primedivine (talk) 15:18, 31 May 2016 (UTC)

That paper calls the period the Wall number, but otherwise it appears to be the same thing. In other sources, this is sometimes also called the Fibonacci entry point. Using Pisano period has the advantage that there is already a Wikipedia article about it that we can link to. Sławomir
Biały 16:09, 31 May 2016 (UTC)

Yes, that would appear to be nearly the same locution, however it was "not proven" that the period would have same property if there were a Wall Sun Sun prime with a great exponent.

The proper terminology should be used as the standard anyways, since it remains the same if the question is answered. Primedivine (talk) 16:18, 31 May 2016 (UTC)

I don't know what you mean. One of several equivalent characterizations of WSS primes is that $\pi (p^{2})=\pi (p)$. Also, I do not think there is a "proper terminology" in this area. It is better to link to the standard term that has its own Wikipedia article. Sławomir
Biały 16:25, 31 May 2016 (UTC)

The pisano period and entry point are two different things. The entry point divides the pisano period, as I described above.

PP=1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24...

EP=1, 3, 4, 6, 5, 12, 8, 6, 12, 15, 10, 12... Primedivine (talk) 16:28, 31 May 2016 (UTC)

OK, I think I understand. In reference 2 of the article, Z(p) is the entry point, and P(p) is the period. The equivalent characterization there is Z(p)=Z(p^2). Sławomir
Biały 16:39, 31 May 2016 (UTC)

Yes, and Z(p) is not exactly standard either, but there are a few other terms that are the same as entry point, like "Rank of apparition" and "restricted period" I believe. Primedivine (talk) 16:51, 31 May 2016 (UTC)

Wolfram doesn't mention WSS in the Pisano article.

It is stated here with justification http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#section6.5

"...the Pisano period mod n, may not be the first Fibonacci number which has n as a factor.
For example the Fibonacci numbers mod 3 have a cycle of length 8, so that Fib(8)=Fib(0) mod 3, and in general Fib(n)=Fib(n+8) mod 3."

After careful review I have concluded that the origin of the conjecture's definition is in the Sun Sun paper. It refers to the restricted period not the pisano period: Remark 5, page 385 http://matwbn.icm.edu.pl/ksiazki/aa/aa60/aa6046.pdf

The 1992 paper explicitly states n(p) as $\alpha (p)$, the least positive subscript such that p | F(n(p)). The work of Robinson and Vinson in 1963 had already stated Wall's theorems clearly. For example $\alpha (p)$ appears in Robinson's paper page 30 here: http://www.fq.math.ca/Scanned/1-2/robinson.pdf. Marc Renault has published recently using this notation of Robinson's, and he also has done great work to standardize the notation. See here: http://webspace.ship.edu/msrenault/fibonacci/fib.htm

In Vinson's paper page 1 and page 41 we see f(m) as the restricted period alpha with the same proof point using the lcm property originating from Wall's paper as d(m) theorem 2-3 page 526. http://www.fq.math.ca/Scanned/1-2/vinson.pdf

Primedivine (talk) 01:26, 4 June 2016 (UTC)

Non-existence hypothesized

The revert was due to Wall's remark in Theorem 5 of his 1960 paper (emphasis added):

The most perplexing problem we have met in this study concerns the hypothesis $k(p^{2})\neq k(p)$. We have run a test on digital computer which shows that $k(p^{2})\neq k(p)$ for all p up to 10000; however, we cannot prove that $k(p^{2})=k(p)$ is impossible. The question is closely related to another one, "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, one might conjecture that equality may hold for some exceptional p.

— D.D. Wall, Fibonacci Series Modulo m, The American Mathematical Monthly, pp. 528, Theorem 5

Note that he tested the original hypothesis, and could not find a counterexample. Primedivine (talk) 12:59, 17 November 2016 (UTC)

01:01, 18 November 2016‎ Eric Rowland (talk | contribs)‎ . . (13,145 bytes) (-127)‎ . . (No, read the whole remark. If anything, he hypothesized that they do exist: "one might conjecture that equality may hold for some exceptional p"

His paper is based upon a strong hypothesis, not the weaker conjecture he mentioned only once with no follow up, although as a perplexing open problem with a lack of proof of the hypothesis, he is conjecturing both no matter what his preference actually was. That cannot be argued. Primedivine (talk) 15:15, 19 November 2016 (UTC)

When a mathematician makes a conjecture, it's a clear and unambiguous statement about what he or she guesses is true, and it's usually labeled with a "Conjecture." heading. Wall didn't do that one way or the other about the existence of Wall–Sun–Sun primes. He might be saying that the evidence suggests they don't exist, but then he quickly points out that a related question does have examples and so there may be "exceptional" Wall–Sun–Sun primes too. The "hypothesis" is the hypothesis of his Theorem 5; this is simply part of an "if ... then ..." statement and not a conjecture. Eric Rowland (talk) 18:53, 19 November 2016 (UTC)

You can't possibly be a native English speaker. A conjecture is proposition that may be based on inconclusive grounds, or incomplete information, and sometimes can not be fully tested. The hypothesis in this case was not meant to be "if/then", it was meant as a testable conjecture based on clearly accepted grounds, that is clearly stated throughout the paper. He is an applied scientist, at the time working for IBM. You clearly have not read the paper, and know nothing about him. In fact, the passing remark at the end, "We note in passing that the number of ordered pairs (a,b) (mod m) with (a,b,m)=1 is give by the formula m^2 Prod p|m (1-1/p^2)." In English, a passing remark is a brief statement without much merit, that has been made as a side note off of the main topic/subject. His intent is absolutely clear, in passing. Primedivine (talk) 06:19, 20 November 2016 (UTC)

Trying to guess what his intent is "in passing" is extremely problematic because you can't cite it. You or I can guess what Wall believed all we want, but his paper is the only source material you've cited, and the paper doesn't contain a conjecture. If you think it does, please quote the statement where he makes the conjecture — not a paragraph where he discusses what may or may not be true without taking a stance, but a single statement where he writes "One conjectures that ..." or "I suspect that ..." or "It seems highly likely that ..." or similar. In the meantime, stop promoting misinformation. Without such a statement, no mathematician would call this a conjecture. Being unable to prove that $k(p^{2})=k(p)$ is impossible is not the same as conjecturing that it's impossible. As far as I can tell, the closest he comes to a conjecture is the statement "one might conjecture that equality may hold for some exceptional $p$", which is equivalent to "one might conjecture that a Wall–Sun–Sun prime exists"; he's still not making a conjecture here (otherwise he would have omitted "might"), but even if he were this is the opposite of what you're claiming he conjectured. Eric Rowland (talk) 17:47, 20 November 2016 (UTC)

So now you are just saying the opposite of anything sane, but then back peddle, and swap your arguments at random. You reversed your view, now you say he did not conjecture they exist. Good, I got through to you a little. I was the one staying neutral the whole time, and finally offering a personal opinion that has nothing to do with my edit. You on the other hand are pressing your opinion to twist the words of the author. You seem to think, that my edit means something that it doesn't. I can tell you have not read the paper, so you are spreading misinformation. That fact that he made a scientific hypothesis is not up for debate, and you know it. That is why you changed your position. How is a the whole Theorem 5, a hypothesis if/then? Explain that. It seems you need to cite something more specific to substantiate your claim that he did not mean a testable scientific hypothesis. One can hypothesize something without personal feelings, in fact that is just what he did. The hypothesis was based upon observations, and the scientific process including clear mathematical attempts to prove that they are impossible, or not.

The paper starts with these key sentences as an open question: "This inquiry is concerned with determining the length of the period ...The problem 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 questions remain unanswered: see remarks after theorems 5 and 7." Actually, the paper is laid out with Theorems 3-7 based upon 1-2 as the foundation, all focused on the same attempt, to prove that they are impossible, ie on page 526 "In view of theorem 2(lcm property which would be an attempt to squeeze prove they were impossible by lcm of the indices, which almost works except for prime powers)... The next five theorems establish properties of the special case series.." Then Theorems 8 through, follow up successively. If you have read the paper, then tell me what Theorem 7+ or above is about? What is the remark in Theorem 7 about? A well stated well tested open problem like this(with a clear mathematical path of either one or the other) is automatically a weak conjecture, although he didn't officially conjecture anything, as I've said from the beginning. I know many professional mathematicians that agree with that, so unless you have outstanding credentials beyond that of the mathematicians with published work in the field, then the revert stands. He met the problem concerning the hypothesis as he stated. So far I've posted all the quotes which clearly state what I paraphrased in the edit. You have presented nothing so far to support your revert. Primedivine (talk) 08:39, 21 November 2016 (UTC)

BTW, next time follow proper procedure and use the talk page before making multiple reverts without substantiation. I hadn't even seen the quoted revert, without a notification. The quote is perfect though. Primedivine (talk) 08:39, 21 November 2016 (UTC)

History

The origin of the name cited by Z. Sun came from R. Crandall, K. Dilcher, and C. Pomerance in 1997.

In the absence of future theoretical results, one is moved to attach the probability 1/p also to certain other properties (mod p^2). What might be called Wall-Sun-Sun primes

— Richard Crandall,Karl Dilcher, and Carl Pomerancel, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), pp. 447

Primedivine (talk) 13:27, 17 November 2016 (UTC)

Unexplained / Unreferenced jargon

In the second sentence, the terms "period" and "modulo prime" are used and not explained. They are meaningless to a non mathematician and need to be explained. — Preceding unsigned comment added by 2A02:C7F:C409:DA00:B431:7BFA:864E:7D5B (talk) 17:48, 17 May 2017 (UTC)

I added some links. Pisano period has more info. Eric Rowland (talk) 00:55, 20 June 2017 (UTC)

Legendre symbols

The notation I learned for Legendre symbols is with the parentheses and the p on 5 as shown in this article, but without the fraction bar. I think this is pretty standard, and there's the LaTeX \legendre that generates the symbol I'm used to. Should this be updated in the article here? Joshuazucker (talk) 00:00, 23 December 2023 (UTC)

Never mind, it looks like people have done them with various bars (dashed, or thicker than a fraction bar) or lack thereof, and the usage here is pretty standard. It should stay as it is. And \legendre was just a user-defined thing in some local dialects of LaTeX. Oops. Joshuazucker (talk) 00:05, 23 December 2023 (UTC)

Tables of WSS primes and adjacent concepts

Hi I used to use this page quite a lot, and there were some very useful large tables on it which have recently been removed. Are they still accessible somewhere else, please? I found them extremely helpful in checking my own work on this topic. Thank you. Geemah (talk) 20:46, 4 May 2024 (UTC)

Anything that is deleted is still accessible via the history, in this case, you might want to try [[1]] It's probably unfortunate for you that David Eppstein decided to eliminate the tables, probably because it looked like original research as it was unsourced. You could try to restore the tables (but it might be reverted), or move them to your own space, and update them there. Dhrm77 (talk) 14:20, 16 May 2024 (UTC)

Aha! Thank you very much @Dhrm77: I am very new to this and did not know of this possibility. If @David Eppstein is reading this and knows of a(n alternative) source of the data in the tables then I would be happy to collate it and put it back up with appropriate references etc -- at least I'd learn how to use the editor! My main problem is I never knew where it came from in the first place. Thanks again. Geemah (talk) 14:33, 17 May 2024 (UTC)

I removed the tables because there is a persistent and globally-banned editor who has been filling our number theory articles with pages after pages of cruft like this that in some cases can be sourced to and duplicates OEIS and in some cases of original research. The same editor has also created huge numbers of sockpuppets to reinstate the same content. Beyond the fact that it makes our articles unreadable and duplicates OEIS, WP:DENY is also a reason for removing this content. Please do not put it back. —David Eppstein (talk) 17:32, 17 May 2024 (UTC)

Thank you -- I have learned something new (including the word "cruft"!). I am still trying to find something reliable in the literature on these things but the OEIS definitely only contains a tiny amount of the information purportedly contained in those deleted tables.

But I take the point -- I will take the (historical version of the) tables with a grain of salt (obviously I always verified them anyway but they always seemed to me to be accurate and complete within the stated bounds) -- and I will find some way of making that content available, once it is more robustly formulated, which does not involve "cruft" ... Geemah (talk) 14:21, 18 May 2024 (UTC)

Wall-Sun-Sun primes in Lucas sequences

Let T(P,Q,n) be the n-th term of the Lucas sequence (P,Q) of the first kind, then we have $T(k,1,n)={\frac {T({\sqrt {k-2}},-1,2n)}{\sqrt {k-2}}}$ for k ≥ 3. For odd primes p that are not factors of k² - 4 (so that p - (⁠k² - 4/p⁠) is even), we always have $p\mid {\dfrac {T\left({\sqrt {k-2}},-1,p-\left({\tfrac {k^{2}-4}{p}}\right)\right)}{\sqrt {k-2}}}=T\left(k,1,{\dfrac {p-\left({\tfrac {k^{2}-4}{p}}\right)}{2}}\right)$, so it would be meaningful to ask when p² divides this quantity. The n-Wall-Sun-Sun primes correspond to k = n² + 2.

Note the product formula ${\frac {T({\sqrt {k-2}},-1,n)}{\sqrt {k-2}}}=\prod _{d|n,d\geq 3}P_{d}(k)$ for even n and $T({\sqrt {k-2}},-1,n)=\prod _{d|n,d\geq 3}P_{d}(k)$ for odd n, where P_n is the minimal polynomial of -2 cos ⁠2π/n⁠.

The OEIS has A238490 for k = 4, A238736 for k = 6 and A337791 for k = 38. 129.104.241.193 (talk) 09:25, 15 May 2024 (UTC)