On the supercongruence conjectures of van Hamme

Holly Swisher

doi:10.1186/s40687-015-0037-6

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.

Highlights

In 1914, Ramanujan listed 17 infinite series representations of 1/π, including for example ∞ (4k + 1)(−1)k ( )3k k!3 k=0 = 2 π ΓSeveral of Ramanujan’s formulas relate hypergeometric series to values of the gamma function.In the 1980’s it was discovered that Ramanujan’s formulas provided efficient means for calculating digits of π
Nebe prove a general p−adic analog of Ramanujan type supercongruences modulo p2 for suitable truncated hypergeometric series arising from complex multiplication (CM) elliptic curves
The supercongruence (B.2) has been proved in three ways, by Mortenson [13] using a technical evaluation of a quotient of Gamma functions, by Zudilin [18] using the W-Z method, and by Long [9] using hypergeometric series identities and evaluations

Summary

Introduction

In 1914, Ramanujan listed 17 infinite series representations of 1/π, including for example. Nebe prove a general p−adic analog of Ramanujan type supercongruences modulo p2 for suitable truncated hypergeometric series arising from CM elliptic curves. The supercongruence (B.2) has been proved in three ways, by Mortenson [13] using a technical evaluation of a quotient of Gamma functions, by Zudilin [18] using the W-Z method, and by Long [9] using hypergeometric series identities and evaluations. (D.2) has been proved by Long and Ramakrishna in a recent preprint [10] They prove that (H.2) holds modulo p3 when p ≡ 1 (mod 4), and provide extensions for (D.2) and (H.2) to additional primes. For primes p ≡ 2 (mod 3), Theorem 1.1 yields the following new generalization of (E.2). Since this case corresponds to a CM elliptic curve, we know that (K.2) holds modulo p2 by [4]; it remains to be proved modulo p4 as conjectured

Preliminaries

Conjectures