Wall-Sun-Sun primes and Fibonacci divisibility

Importance: Medium ✭✭
Author(s):
Keywords: Fibonacci
prime
Recomm. for undergrads: no
Posted by: adudzik
on: June 14th, 2008
Conjecture   For any prime $ p $, there exists a Fibonacci number divisible by $ p $ exactly once.

Equivalently:

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.

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 Wall-Sun-Sun primes.

Bibliography

[EJ] Andreas-Stephan Elsenhansand and Jörg Jahnel, The Fibonacci sequence modulo p^2

[R] Marc Renault, Properties of the Fibonacci Sequence Under Various Moduli

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


* indicates original appearance(s) of problem.