Congruences for the Apéry numbers modulo p³

2024/09/10 by Zhi-Hong Sun, Sun, Zhi-Hong
Mathematics · #05A19 #11B68 #11E25 #Advanced Mathematical Identities #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #Secondary 05A10

paper · pdf · doi:10.48550/arxiv.2409.06544

openalex publication_date 2024/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \A'_n\ be the Apéry numbers given by A'_n=∑_k=0ⁿ\binom nk²\binomn+kk. For any prime p≡ 3\pmod 4 we show that A'_\fracp-12≡ \fracp²3\binom\fracp-32\fracp-34^-2\pmod p³. Let \t_n\ be given by t₀=1, t₁=5 \hboxand t_n+1=(8n²+12n+5)t_n-4n²(2n+1)²t_n-1 (n≥ 1). We also obtain the congruences for t_p\pmod p³, t_p-1\pmod p² and t_\fracp-12\pmod p², where p is an odd prime.

Related

• New conjectures involving binomial coefficients and Apéry-like numbers
• Super congruences concerning binomial coefficients and Apéry-like numbers
• More congruences from Apery-like formulae
• New congruences for sums involving Apery numbers or central Delannoy numbers
• Note on super congruences modulo p²