On the supercongruence conjectures of van Hamme

By Holly Swisher. 2015

166 citations recorded by the source corpus.

In 1997, van Hamme developed $$p$$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 $$p$$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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/5eca91a30c1e22a5d141f8d035602315bad6166f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

On Van Hamme's (A.2) and (H.2) supercongruences (2018, methodology): “[12] H. Swisher, On the supercongruence conjectures of van Hamme, Res.”
On the (K.2) supercongruence of Van Hamme (2015, methodology): “Recently, this strategy has been Date: 8 April 2015.”
Some congruences involving fourth powers of central q-binomial coefficients (2019, background): “Swisher [21] has made many conjectural supercongruences that generalize van Hamme’s 13 Ramanujan type supercongruences.”
Hypergeometric Functions over Finite Fields (2015, methodology): “See [20] for a general result, and [63] by the second and third authors, [85] by the fourth author, [71] by Osburn and Zudilin for some recent developments on Ramanujan-type supercongruences.”
A q-microscope for supercongruences (2018, methodology): “Development of methods [7, 20–22, 24, 25, 27, 31, 32, 34] for establishing Ramanujan-type supercongruences like (2) and (3), sometimes modulo a smaller power of p and normally on a case-by-case study, was mainly hypergeometric.”
Some congruences related to a congruence of Van Hamme (2019, methodology): “For more supercongruences and q-supercongruences, we refer the reader to [4, 6–20, 26, 29, 30].”
A family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial (2019, background): “It should be pointed out that all of the 13 supercongruences have been proved by different techniques (see [27,31]).”