Congruences involving binomial coefficients and Lucas sequences

Abstract

In this paper we obtain some congruences involving central binomial coefficients and Lucas sequences. For example, we show that if p>5 is a prime then ${\sum }_{k=0}^{p-1}{F}_k\ast binom(2k,k)/12^k$ is congruent to 0,1,-1 modulo p according as p=1,4 (mod 5), p=13,17 (mod 30), and p=7,23 (mod 30) respectively, where {F_n} is the Fibonacci sequence. We also raise several conjectures.

23 pages. The current Th. 1.4 is new

References in corpus (6)

• On congruences related to central binomial coefficients
• An elementary proof of a Rodriguez-Villegas supercongruence
• 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

Cited by in corpus (3)

• Open Conjectures on Congruences
• New congruences involving products of two binomial coefficients
• Curious congruences for Fibonacci numbers