Congruences involving binomial coefficients and Lucas sequences

2009/12/07 by Zhi‐Wei Sun, Sun, Zhi-Wei · 1 citation
Mathematics · Physics and Astronomy · #05A10 #11A07 #11B39 #11B65 #11E25. #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0912.1280

openalex publication_date 2009/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∑_k=0^p-1F_k*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.

Cited by

• Curious congruences for Fibonacci numbers

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