{"paper":{"title":"On $p$-adic congruences involving $\\sqrt d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bo Jiang, Zhi-Wei Sun","submitted_at":"2025-04-16T16:47:10Z","abstract_excerpt":"Let $p$ be an odd prime and let $d$ be an integer not divisible by $p$. We prove that $$ \\prod_{1\\le m,n\\le p-1\\atop p\\nmid m^2-dn^2}\\ (x-(m+n\\sqrt{d})) \\equiv \\begin{cases}\\sum_{k=1}^{p-2}\\frac{k(k+1)}2x^{(k-1)(p-1)}\\pmod p &\\text{if}\\ (\\frac dp)=1,\\\\\\sum_{k=0}^{(p-1)/2}x^{2k(p-1)} \\pmod p&\\text {if}\\ (\\frac dp)=-1, \\end{cases}$$ where $(\\frac dp)$ denotes the Legendre symbol. This extends a recent conjecture of N. Kalinin. We also obtain the Wolstenholme-type congruence $$\\sum_{1\\le m,n\\le p-1\\atop p\\nmid m^2-dn^2}\\ \\ \\frac1{m+n\\sqrt d}\\equiv0\\pmod{p^2}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.12242","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.12242/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}