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arXiv · 2111.08778

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

Guo-Shuai Mao
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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.

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Guo-Shuai Mao. 2021-11-16. Proof of some congruence conjectures of Z.-H. Sun involving Ap\'{e}ry-like numbers. https://doi.org/10.1080/10236198.2023.2234048

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