Talk:Pisano period

Zeros

Has it been proven that there are always only 1,2,4 zeroes in a sequence? —Preceding unsigned comment added by -Ozone- (talk • contribs) 2 January 2006.

Yes, I added the reference.--Patrick (talk) 13:30, 21 December 2007 (UTC)

Musicality ?

http://code.google.com/p/pisanomatic/ suggests a link to music ... --195.137.93.171 (talk) 11:45, 29 August 2010 (UTC)

History

The article is missing a history section. Who first proposed Pisano periods?--♦IanMacM♦ ^{(talk to me)} 08:01, 21 September 2011 (UTC)

The MathWorld article and this book suggest that Joseph Louis Lagrange was the first person to note this property of Fibonacci numbers. Would it be OK to put this in the article?--♦IanMacM♦ ^{(talk to me)} 09:26, 21 September 2011 (UTC)

Restructuring?

This article is not very easy to read. There is lots of information on Pisano periods and their generalisations available, but this article does not present it in a systematic and helpful manner. There are many tables that are not particularly illuminating, and sections that have no real motivation. What is the point of the sums section? The powers of 10 section is completely unnecessary, and betrays a lack of mathematical sophistication: the fact about the Pisano periods of 10^n is very simple to show. To claim that it was 'proved' by a specific person does not accurately reflect the culture of mathematics.

Is anybody watching this article? I'd like to have a go at rewriting it, but I don't want to step on anyone's toes. On the other hand, I've been thinking a lot about Pisano periods recently, and this article has been of no help at all. I feel strongly that it can be made better.

Robodile (talk) 23:34, 3 January 2016 (UTC)

Wall-Sun-Sun primes question

The article currently states: "Any prime p providing a counterexample would necessarily be a Wall-Sun-Sun prime, and such primes are also conjectured not to exist." However in the Wall–Sun–Sun prime article the opposite is stated: "In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known."

TraxPlayer (talk) 21:25, 15 February 2016 (UTC)

This is not contradictory. In fact, from the article Wall–Sun–Sun prime, one may see that that two opposite conjectures have been set, the non-existence, as well as the existence of infinitely many Wall-Sun Sun primes. This means that some mathematicians believe one thing, and some believe the opposite. As a belief is not a proof the two assertions that you have quoted are true and not contradictory. D.Lazard (talk) 09:23, 16 February 2016 (UTC)

To dos for rewrite.

Major:
One large generalisations section, with fewer lists of numbers.
Algebraic number theory section addresses the more general case of (a,b)-Fibonacci numbers.

Minor:
Alphabetize references.
Remove 'cultural references'? It's just trivia really. Would be better suited to TV tropes.

Questions:
Is the 'properties' section now too detailed? Perhaps it is better not to include details of the proof that the periods divide p-1 or 2(p+1) for non-anomolous primes. Should the reader just be sent to a reference?

Robodile (talk) 11:37, 1 February 2016 (UTC)

About the "properties section": It is not too long, as many articles, including good articles have longer sections. MOS:MATH#Proofs says we often want to include proofs, as a way of really exposing the meaning of some theorem, definition, etc. This is the case here: without sketched proofs, the results of this section, (particularly the final general upper bound on the Pisano periods) appear as "miracles", and thus may appear as dubious to the reader who does not want to access the sources. Thus, giving information explaining how these results are natural consequences of standard theorems is important, including from encyclopedic point of view. Presently, the proofs are far to be complete or detailed. I have tried to keep them as short as possible, while allowing an experimented mathematician to retrieve them rather easily.

About the case distinction for non-anomolous primes: I came to it by examining the table of the next section, and trying to understand why some primes have a different behavior. This is for explaining these surprising behavior that I have introduced the proofs. Another surprise that needed explanation, is that there are periods of 2(p+1) and 2(p+1)/3, but not 2(p+1)/2. In other words, if it is possible that a reader remarks that the tables suggest some properties, then these properties must be explained in the text. That is what I have tried to do when sketching the proofs.

About the other sections: I agree that a large work of rewriting is needed, however, presently, I have not the time to do that. D.Lazard (talk) 14:25, 1 February 2016 (UTC)

About the proofs: Okay, I agree with you that the properties section is not too long. I'll probably still do some tinkering with the presentation of the section, perhaps to better distinguish the properties from the proofs, but I doubt I'll change the proofs themselves. It's probably also useful to have them here, because the results in the algebraic number theory section are analogous, but it would be over-detailed to include them in full generality there. Anyone sufficiently interested in Pisano periods and algebraic number theory should be able to generalise to the general case from the proofs in the specific case.

About the rest of the article: Yes, there is a lot of rewriting to do. I decided to take on this article as I wanted a project to learn how to improve wikipedia articles, and I had also been playing with Pisano periods in some research recently. I'm happy to rewrite the rest of the article slowly in the coming weeks. You've been helpful keeping an eye on things so far, so any comments or changes you want to make are most welcome.Robodile (talk) 15:46, 3 February 2016 (UTC)

There is a proof that is still lacking in property section, that is the proof that π(p^k) = p^sπ(p) for s ≤ k (in fact the article states a weaker result). If you know a short way for indicating how this can be proven, it would be useful to add it. D.Lazard (talk) 16:34, 3 February 2016 (UTC)

Refinement required

If p is a prime different from 2 and 5, then the modulo p analogue of Binet's formula implies that π(p) is the multiplicative order of the roots of x² – x – 1 modulo p.

Above statement must be refined. Taking p=11, the roots are 4 (multiplicative order 5, as shown by the sequence (1,4,5,9,3(,1)) and 8 (multiplicative order 10). Leen Droogendijk (talk) 11:26, 13 June 2020 (UTC)