Three supercongruences for Apery numbers or Franel numbers

2021/03/10 by Yong Zhang, Zhang, Yong
Mathematics · #11B39 #11B65 #11B75 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #Secondary 05A10

paper · pdf · doi:10.48550/arxiv.2103.05830

openalex publication_date 2021/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Apéry numbers A_n and the Franel numbers f_n are defined by A_n=∑_k=0ⁿ\binomn+k2k²\binom2kk² \rm and f_n=∑_k=0ⁿ\binomnk³(n=0, 1, ⋯,). In this paper, we prove three supercongruences for Apéry numbers or Franel numbers conjectured by Z.-W. Sun. Let p≥ 5 be a prime and let n∈ ℤ⁺. We show that (1)/(n)(∑_k=0^pn-1(2k+1)A_k-p∑_k=0^n-1(2k+1)A_k)≡0\pmodp^4+3ν_p(n) and (1)/(n³)(∑_k=0^pn-1(2k+1)³A_k-p³∑_k=0^n-1(2k+1)³A_k)≡0\pmodp^6+3ν_p(n), where ν_p(n) denotes the p-adic order of n. Also, for any prime p we have (1)/(n³)(∑_k=0^pn-1(3k+2)(-1)^kf_k-p²∑_k=0^n-1(3k+2)(-1)^kf_k)≡0\pmodp³.

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• Proof of a supercongruence conjectured by Z.-H. Sun
• Supercongruences via Beukers' method
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• Supercongruences involving binomial coefficients and Euler polynomials
• Congruences involving Franel and Catalan-Larcombe-French numbers