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arXiv · 2508.00343

Streamlined WZ method proofs of Van Hamme supercongruences

Andres Valloud
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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 $\Gamma_p$ 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 $p^4$, and of H.2 modulo $p^3$ 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.

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Andres Valloud. 2025-08-01. Streamlined WZ method proofs of Van Hamme supercongruences. https://doi.org/10.1007/s11139-025-01232-1

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