Open Access

On the supercongruence conjectures of van Hamme

Research in the Mathematical Sciences20152:18

https://doi.org/10.1186/s40687-015-0037-6

Received: 10 April 2015

Accepted: 3 August 2015

Published: 15 October 2015

Abstract

In 1997, van Hamme developed \(p\)–adic analogs, for primes p, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the \(p\)–adic gamma function, and are called Ramanujan-type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the proved ones, and provide some extensions.

Keywords

Ramanujan-type supercongruences Hypergeometric series