Supercongruences for sums involving Domb numbers

TL;DR

This paper proves supercongruences involving Domb numbers, confirming conjectures by Sun, and relates finite sums modulo primes to classical formulas for 1/π, advancing understanding of number theory and special sequences.

Contribution

It establishes new supercongruences for sums involving Domb numbers, confirming four conjectures of Sun and connecting finite sums to classical π-related formulas.

Findings

Proves supercongruences for sums involving Domb numbers.

Confirms four conjectures of Z.-W. Sun.

Links finite prime sums to Rogers' formula for 1/π.

Abstract

We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any prime $p\ge 5$, \begin{align*} \sum_{k=0}^{p-1}\frac{3k+1}{(-32)^k}{\rm Domb}(k)\equiv (-1)^{\frac{p-1}{2}}p+p^3E_{p-3} \pmod{p^4}, \end{align*} which is regarded as a $p$-adic analogue of the following interesting formula for $1/\pi$ due to Rogers: \begin{align*} \sum_{k=0}^{\infty}\frac{3k+1}{(-32)^k}{\rm Domb}(k)=\frac{2}{\pi}. \end{align*} Here ${\rm Domb}(n)$ and $E_n$ are the famous Domb numbers and Euler numbers.