Streamlined WZ method proofs of Van Hamme supercongruences
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908818771804160 |
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| author | Valloud, Andres |
| author_facet | Valloud, Andres |
| contents | 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 $p^4$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00343 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Streamlined WZ method proofs of Van Hamme supercongruences Valloud, Andres Number Theory 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 $p^4$, 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. |
| title | Streamlined WZ method proofs of Van Hamme supercongruences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.00343 |