A refinement of a congruence result by van Hamme and Mortenson

TL;DR

This paper refines a known congruence involving binomial sums by extending it to higher powers of primes and introduces new congruences related to Euler numbers and binomial coefficients.

Contribution

It extends van Hamme and Mortenson's congruence to modulo p^4 and establishes new congruences involving Euler numbers and binomial coefficients.

Findings

Extended the congruence to modulo p^4 involving Euler numbers.

Proved a new congruence involving binomial coefficients and powers of 2.

Provided explicit formulas for sums related to prime moduli.

Abstract

Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\sum_{k=0}^{(p-1)/2}(4k+1)\binom{-1/2}k^3\equiv (-1)^{(p-1)/2}p\pmod{p^3}.$$ In this paper we show further that \begin{align*}\sum_{k=0}^{p-1}(4k+1)\binom{-1/2}k^3\equiv &\sum_{k=0}^{(p-1)/2}(4k+1)\binom{-1/2}k^3 \\\equiv & (-1)^{(p-1)/2}p+p^3E_{p-3} \pmod{p^4},\end{align*}where $E_0,E_1,E_2,\ldots$ are Euler numbers. We also prove that if $p>3$ then $$\sum_{k=0}^{(p-1)/2}\frac{20k+3}{(-2^{10})^k}\binom{4k}{k,k,k,k}\equiv(-1)^{(p-1)/2}p(2^{p-1}+2-(2^{p-1}-1)^2)\pmod{p^4}.$$