Proof of some congruence conjectures of Z.-H. Sun involving Ap\'{e}ry-like numbers

TL;DR

This paper proves a conjecture by Z.-H. Sun involving congruences of sums with Apéry-like numbers, connecting binomial coefficients, Franel numbers, and Euler numbers modulo prime powers.

Contribution

The paper establishes a proof for a specific congruence conjecture involving Apéry-like numbers and related special sequences, advancing understanding of their modular properties.

Findings

Confirmed the conjecture for primes p > 3

Derived congruences involving binomial, Franel, and Euler numbers

Enhanced knowledge of number theoretic properties of Apéry-like sequences

Abstract

In this paper, we mainly prove the following conjecture of Z.-H. Sun cite{SH20}: Let $p>3$ be a prime. Then $$\sum_{k=0}^{p-1}\binom{2k}k\frac{3k+1}{(-16)^k}f_k\equiv(-1)^{(p-1)/2}p+p^3E_{p-3}\pmod{p^4},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ and $E_n$ stand for the $n$th Franel number and $n$th Euler number respectively.