Streamlined WZ method proofs of Van Hamme supercongruences

2025/08/01 by Valloud, Andres
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.00343

Abstract

Using the WZ method to prove supercongruences critically depends on an inspired WZ pair choice. This paper demonstrates a procedure for finding WZ pair candidates to prove a given supercongruence. When suitable WZ pairs are thus obtained, coupling them with the p-adic approximation of Γ_p by Long and Ramakrishna enables uniform proofs for the Van Hamme supercongruences B.2, C.2, D.2, E.2, F.2, G.2, and H.2. This approach also yields the known extensions of G.2 modulo p⁴, and of H.2 modulo p³ when p is 3 modulo 4. Finally, the Van Hamme supercongruence I.2 is shown to be a special case of the WZ method where Gosper's algorithm itself succeeds.

Citations

• Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher
• Supercongruences using modular forms
• q-analogues of the G.2 Supercongruence of Van Hamme
• On van Hamme's (A.2) and (H.2) supercongruences
• A q-microscope for supercongruences
• On the supercongruence conjectures of van Hamme
• Some supercongruences occurring in truncated hypergeometric series
• Some supercongruences occurring in truncated hypergeometric series
• Hypergeometric evaluation identities and supercongruences
• Open Conjectures on Congruences

Related

• Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher
• On the supercongruence conjectures of van Hamme
• q-Analogues of Dwork-type supercongruences
• A refined q-analogue of some congruences of Van Hamme
• q-analogues of the G.2 Supercongruence of Van Hamme