Streamlined WZ method proofs of Van Hamme supercongruences

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发表日期：

2025年11月5日

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Andres Valloud

摘要

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 $\Gamma _p$ $Γ_{p}$ 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 $p^4$ $p^{4}$ , and of (H.2) modulo $p^3$ $p^{3}$ 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.