Curious congruences for Fibonacci numbers

TL;DR

This paper derives new congruences involving Fibonacci numbers and binomial coefficients modulo prime squares, expanding understanding of their number-theoretic properties and proposing related conjectures.

Contribution

It establishes novel congruences for Fibonacci-related sums modulo prime squares and explores similar results for other second-order recurrences.

Findings

Derived congruences for sums involving Fibonacci numbers and binomial coefficients modulo p^2.

Extended results to other second-order recurrence sequences.

Proposed several new conjectures in the area of Fibonacci number congruences.

Abstract

In this paper we establish some sophisticated congruences involving central binomial coefficients and Fibonacci numbers. For example, we show that if $p\not=2,5$ is a prime then $$\sum_{k=0}^{p-1}F_{2k}\binom{2k}{k}=(-1)^{[p/5]}(1-(p/5)) (mod p^2)$$ and $$\sum_{k=0}^{p-1}F_{2k+1}\binom{2k}k=(-1)^{[p/5]}(p/5) (mod p^2).$$ We also obtain similar results for some other second-order recurrences and raise several conjectures.