Curious congruences for Fibonacci numbers

Details Curious congruences for Fibonacci numbers Curious congruences for Fibonacci numbers

2009-12-14

arxiv.org/abs/0912.2671v4

Mathematics

Number Theory

16 pages. Revise Conj. 4.2

Scientific paper

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.

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