On the (K.2) supercongruence of Van Hamme

TL;DR

This paper proves the final case of Van Hamme's 13 supercongruence conjectures related to Ramanujan-type identities, using classical congruences and WZ pairs, and discusses future research directions.

Contribution

It completes the proof of all remaining Van Hamme supercongruences, employing classical methods and WZ pairs, advancing understanding of Ramanujan-type supercongruences.

Findings

Proof of the last Van Hamme supercongruence case

Application of classical congruences and WZ pairs

Discussion of future research directions

Abstract

We prove the last remaining case of the original 13 Ramanujan-type supercongruence conjectures due to Van Hamme from 1997. The proof utilizes classical congruences and a WZ pair due to Guillera. Additionally, we mention some future directions concerning this type of supercongruence.