A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$

TL;DR

This paper proves supercongruences related to Almkvist-Zudilin numbers that support conjectures connecting these sums to classical formulas for 1/π, extending Chan and Verrill's work into the p-adic realm.

Contribution

The authors establish new supercongruences for Almkvist-Zudilin numbers, confirming conjectures and providing a p-adic analogue of a known formula for 1/π.

Findings

Proved supercongruences for Almkvist-Zudilin sums

Confirmed conjectures of Zudilin and Sun

Connected p-adic sums to classical 1/π formulas

Abstract

We prove three supercongruences for sums of Almkvist-Zudilin numbers, which confirm some conjectures of Zudilin and Z.-H. Sun. A typical example is the Ramanujan-type supercongruence: \begin{align*} \sum_{k=0}^{p-1} \frac{4k+1}{81^k}\gamma_k \equiv \left(\frac{-3}{p}\right) p\pmod{p^3}, \end{align*} which is corresponding to Chan and Verrill's formula for $1/\pi$: \begin{align*} \sum_{k=0}^\infty \frac{4k+1}{81^k}\gamma_k = \frac{3\sqrt{3}}{2\pi}. \end{align*} Here $\gamma_n$ are the Almkvist-Zudilin numbers.