Wall-Sun-Sun primes and Fibonacci divisibility

Importance: Medium ✭✭
Keywords: Fibonacci
Recomm. for undergrads: no
Posted by: adudzik
on: June 14th, 2008

\begin{conjecture} For any prime $p$, there exists a Fibonacci number divisible by $p$ exactly once. \end{conjecture}


\begin{conjecture} For any prime $p>5$, $p^2$ does not divide $F_{p-\left(\frac p5\right)}$ where $\left(\frac mn\right)$ is the Legendre symbol. \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

Let $p$ be an odd prime, and let $\nu_p(n)$ denote the $p$-adic valuation of $n$. Let $F_{k(p)}$ be the smallest Fibonacci number that is divisible by $p$ (which must exist by a simple counting argument). A well-known result says that $\nu_p(F_n)=0$ unless $k(p)$ divides $n$, and $\nu_p(F_{k(p)m}) = \nu_p(F_{k(p)}) + \nu_p(m)$. This conjecture asserts that $\nu_p(F_{k(p)})=1$ for all $p$. This has been verified up to at least $p<10^{14}$. [EJ]

This conjecture is equivalent to non-existence of \href[Wall-Sun-Sun primes]{http://en.wikipedia.org/wiki/Wall-Sun-Sun_prime}.


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

[EJ] Andreas-Stephan Elsenhansand and Jörg Jahnel, \href[The Fibonacci sequence modulo p^2]{http://www.uni-math.gwdg.de/tschinkel/gauss/Fibon.pdf}

[R] Marc Renault, \href[Properties of the Fibonacci Sequence Under Various Moduli]{http://www.math.temple.edu/~renault/fibonacci/thesis.html}

*[W] D. D. Wall, Fibonacci Series Modulo m, American Mathematical Monthly, 67 (1960), pp. 525-532.

* indicates original appearance(s) of problem.