On van Hamme's (A.2) and (H.2) supercongruences

TL;DR

This paper extends certain Ramanujan-type supercongruences conjectured by van Hamme to higher powers of primes, revealing new supercongruences modulo p^4 for specific primes.

Contribution

It introduces new supercongruences modulo p^4 for primes congruent to 3 mod 4, expanding the known results on van Hamme's conjectures using combinatorial identities.

Findings

Extended (A.2) and (H.2) supercongruences to modulo p^4

Identified new supercongruences for primes p ≡ 3 mod 4

Utilized combinatorial identities to prove these results

Abstract

In 1997, van Hamme conjectured 13 Ramanujan-type supercongruences labeled (A.2)--(M.2). Using some combinatorial identities discovered by Sigma, we extend (A.2) and (H.2) to supercongruences modulo $p^4$ for primes $p\equiv 3\pmod{4}$, which appear to be new.