Curious congruences for Fibonacci numbers

Abstract

In this paper we establish some sophisticated congruences involving central binomial coefficients and Fibonacci numbers. For example, we show that if $p\ne 2,5$ is a prime then $$\sum _{k=0}^{p-1}{F}_{2k}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)=(-1{)}^{[p/5]}(1-(p/5))(mod{p}^2)$$ and $$\sum _{k=0}^{p-1}{F}_{2k+1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)=(-1{)}^{[p/5]}(p/5)(mod{p}^2).$$ We also obtain similar results for some other second-order recurrences and raise several conjectures.

16 pages. Revise Conj. 4.2

References in corpus (6)

• On congruences related to central binomial coefficients
• Various congruences involving binomial coefficients and higher-order Catalan numbers
• On sums of binomial coefficients modulo p^2
• p-adic valuations of some sums of multinomial coefficients
• Fibonacci numbers modulo cubes of primes
• Congruences involving binomial coefficients and Lucas sequences

Cited by in corpus (1)

• Open Conjectures on Congruences