Proof of some conjectural congruences involving Domb numbers

TL;DR

This paper proves conjectural congruences involving Domb numbers and prime numbers, extending number theory knowledge on supercongruences and binomial coefficient identities.

Contribution

It confirms conjectures by Z.-H. Sun relating Domb number sums to prime representations and modular arithmetic, providing new supercongruence proofs.

Findings

Established congruences for sums involving Domb numbers modulo p^3.

Connected Domb number sums to prime representations as x^2+3y^2.

Extended understanding of supercongruences in combinatorial number theory.

Abstract

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then we have $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv4x^2-2p-\frac{p^2}{4x^2}\pmod{p^3}, $$ and if $p\equiv2\pmod3$, then $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv-2\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv\frac{p^2}2\binom{\frac{p-1}2}{\frac{p-5}6}^{-2} \pmod{p^3}, $$ where $D_n=\sum_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}$ stands for the $n$th Domb number.