Supercongruences involving Domb numbers and binary quadratic forms

TL;DR

This paper proves two conjectured supercongruences involving Domb numbers and binary quadratic forms, using advanced number theory techniques like Bernoulli polynomials and hypergeometric transformations.

Contribution

It establishes two new supercongruences related to Domb numbers, confirming conjectures and expanding understanding of congruences in number theory.

Findings

Proved supercongruences modulo p^3 for primes greater than 3

Connected Domb numbers with binary quadratic forms

Utilized Bernoulli polynomials and hypergeometric methods

Abstract

In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where $p$ is any prime greater than $3$) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.