{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:OI3UPSJIDOMVTT4RMPVKJXOY4R","short_pith_number":"pith:OI3UPSJI","schema_version":"1.0","canonical_sha256":"723747c9281b9959cf9163eaa4ddd8e44acf85f10c9a6ce854f02d315f30e5ee","source":{"kind":"arxiv","id":"2504.12242","version":1},"attestation_state":"computed","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 $$\\sum_{1\\le m,n\\le p-1\\atop p\\nmid m^2-dn^2}\\ \\ \\frac1{m+n\\sqrt d}\\equiv0\\pmod{p^2}.$$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2504.12242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-04-16T16:47:10Z","cross_cats_sorted":[],"title_canon_sha256":"77bda9d328cd39ce0954360878786980be609e3013ed050a69b978df6af96a80","abstract_canon_sha256":"1cac254bf29edfc8602f523125511d3364980a52b344bdbf8bff0aadb0b41318"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:50:04.738299Z","signature_b64":"bTIfClzy0Dw2Vp86cbV655hfn9nW5QUmgHcollUxuPop+fecuUyXfnbIrpqGOKNsanj1r+FMYWw6LEmOxgDcDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"723747c9281b9959cf9163eaa4ddd8e44acf85f10c9a6ce854f02d315f30e5ee","last_reissued_at":"2026-07-05T10:50:04.737859Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:50:04.737859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2504.12242","created_at":"2026-07-05T10:50:04.737915+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.12242v1","created_at":"2026-07-05T10:50:04.737915+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.12242","created_at":"2026-07-05T10:50:04.737915+00:00"},{"alias_kind":"pith_short_12","alias_value":"OI3UPSJIDOMV","created_at":"2026-07-05T10:50:04.737915+00:00"},{"alias_kind":"pith_short_16","alias_value":"OI3UPSJIDOMVTT4R","created_at":"2026-07-05T10:50:04.737915+00:00"},{"alias_kind":"pith_short_8","alias_value":"OI3UPSJI","created_at":"2026-07-05T10:50:04.737915+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R","json":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R.json","graph_json":"https://pith.science/api/pith-number/OI3UPSJIDOMVTT4RMPVKJXOY4R/graph.json","events_json":"https://pith.science/api/pith-number/OI3UPSJIDOMVTT4RMPVKJXOY4R/events.json","paper":"https://pith.science/paper/OI3UPSJI"},"agent_actions":{"view_html":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R","download_json":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R.json","view_paper":"https://pith.science/paper/OI3UPSJI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.12242&json=true","fetch_graph":"https://pith.science/api/pith-number/OI3UPSJIDOMVTT4RMPVKJXOY4R/graph.json","fetch_events":"https://pith.science/api/pith-number/OI3UPSJIDOMVTT4RMPVKJXOY4R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R/action/storage_attestation","attest_author":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R/action/author_attestation","sign_citation":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R/action/citation_signature","submit_replication":"https://pith.science/pith/OI3UPSJIDOMVTT4RMPVKJXOY4R/action/replication_record"}},"created_at":"2026-07-05T10:50:04.737915+00:00","updated_at":"2026-07-05T10:50:04.737915+00:00"}