Wikipedia:Reference desk/Archives/Mathematics/2015 December 8

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December 8

Can Wall-Sun-Sun primes exist?

I think I understand the Fibonacci-Wieferich, aka Wall-Sun-Sun prime conjecture, however my results are difficult to believe, and yet hard to deny. Can anyone find an error in the following conjecture?

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 $p^{2}$.

$\left\lbrace {n|m}\ {\text{iff}}\ {F_{n}|F_{m}}\right\rbrace$

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

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

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

$example:$

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

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

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

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

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

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

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

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

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

Greater detail on my talk page. Primedivine (talk) 08:16, 8 December 2015 (UTC)

First, we should probably add "We don't referee your original research," to the list of "We don't"s at the top of the page. So I haven't gotten into the details of your claim. But there is a check you should probably do before going further. The article on Wall-Sun-Sun prime's defines an easy generalization called k-Wall-Sun-Sun primes, basically replacing the Fibonacci sequence with a Lucas sequence. The Wall-Sun-Sun primes are then k-Wall-Sun-Sun primes with k=1. So the question is whether your proof can be generalized to show there are no k-Wall-Sun-Sun primes. If so then it must be incorrect since there are k-Wall-Sun-Sun primes known for many values of k>1. So if the proof is valid then you should be able point to the assumption k=1, in other words you should be able to show where the proof breaks down when k>1. This is a good test since Lucas sequences have most of the divisibility properties as the Fibonacci sequence. Presumably the existence of k-WSS primes for so many values of k is the evidenced use to justify the conjecture that 1-WSS primes exist. --RDBury (talk) 17:35, 8 December 2015 (UTC)

Well, we aren't required to referee original research, but I don't think it's a problem if requests of this type come up occasionally. If some of our volunteers choose to help out, they're welcome to - both in my opinion, and also in terms of our current guidelines. SemanticMantis (talk) 17:41, 8 December 2015 (UTC)

Getting off topic, but really we aren't required do anything. Personally I don't have a problem with doing people's homework for them, but only if it's a problem I find interesting, and I suspect the answers I might give would be so terse that they would amount to just hints anyway. We do, though, occasionally get people claiming to have proved the Riemann hypothesis (somehow without knowing anything about complex variables), and/or the twin primes conjecture, etc. I doubt the OP falls into that category but it's probably not a good idea set a precedent saying "Yes, we will find the error in your angle trisection method." --RDBury (talk) 18:17, 8 December 2015 (UTC)

Thank you, I am familiar with k-Wall-Sun-Sun primes and have read up extensively on the subject over the years. I do think it can be shown why it breaks down for k>1, so I will work on that. I'm not sure I have claimed anything quite yet, but I am asking if anyone can see a problem with the two lemmas laid out, that are apparently well known. It's not exactly original research per say if the two parts being discussed are known to be true, It's actually more of a question if I/we understand the parts correctly. An article update would still have to wait for published results.

I don't think my conjecture is that complicated as one would expect a solution to be, but it has a nice twist that could possibly obfuscate researchers, ie n|m iff F(n)|F(m), rather than using the indices to yield the divisibility of Fibonacci numbers, we can use Fibonacci numbers to yield the divisibility of n|m, or more specifically p^2|F(u(p^1)). That was the ah-ha moment for me considering the lcm property. It's clean and easy, so if its wrong it should be easy to point out. I've tried to play devil's advocate against it but could not find a flaw. Primedivine (talk) 19:47, 8 December 2015 (UTC)

The two lemmas stand uncontested by the math help desk then, and so that part can be updated on the related Wikipedia articles. The last three lines above cannot be updated without published results. Primedivine (talk) 06:44, 10 December 2015 (UTC)

Whatever the Wikipedia Mathematics Reference desk says (or doesn't say) about any subject is not in itself considered a reliable source. The Ref desk may well identify a source that can be investigated and, if appropriate, can be cited at an article. -- Jack of Oz ^{[pleasantries]} 06:59, 10 December 2015 (UTC)

"Routine calculations do not count as original research, provided there is consensus among editors that the result of the calculation is obvious, correct, and a meaningful reflection of the sources." The theorem above is pretty clear using only multiplication for the lcm.Primedivine (talk) 11:30, 13 December 2015 (UTC)