{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:CYKXNTKCLM7P4PYTFKYDXOTACG","short_pith_number":"pith:CYKXNTKC","schema_version":"1.0","canonical_sha256":"161576cd425b3efe3f132ab03bba6011b04edbee65e772a4deb8b9da45a9af89","source":{"kind":"arxiv","id":"1610.03384","version":8},"attestation_state":"computed","paper":{"title":"Supercongruences involving Lucas sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2016-10-11T18:17:12Z","abstract_excerpt":"For $A,B\\in\\mathbb Z$, the Lucas sequence $u_n(A,B)\\ (n=0,1,2,\\ldots)$ are defined by $u_0(A,B)=0$, $u_1(A,B)=1$, and $u_{n+1}(A,B) = Au_n(A,B)-Bu_{n-1}(A,B)$ $(n=1,2,3,\\ldots).$ For any odd prime $p$ and positive integer $n$, we establish the new result $$\\frac{u_{pn}(A,B) - (\\frac{A^2-4B}p) u_n(A,B)}{pn} \\in \\mathbb Z_p,$$ where $(\\frac{\\cdot}p)$ is the Legendre symbol and $\\mathbb Z_p$ is the ring of $p$-adic integers.\n  Let $p$ be an odd prime and let $n$ be a positive integer. For any integer $m\\not\\equiv0\\pmod p$, we show that $$\\frac1{pn}\\bigg(\\sum_{k=0}^{pn-1} \\frac{\\binom{2k}k}{m^k} -"},"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":"1610.03384","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-11T18:17:12Z","cross_cats_sorted":[],"title_canon_sha256":"cd2e1cf1f222761104d6b84344fadf851631cb56782fa6e24edf1c3fd08eb497","abstract_canon_sha256":"6babda5cb4875cef29299eda4b81d3bb570ed2ca24eea65b0a29547d1fb3e6bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:59:07.771938Z","signature_b64":"KW0ALDd9a5SoM9zCbx1cCVFjXx0PyLUwMRojO0O13hEbl06F2Bah2Oy3GPqkbABP199lSwYYdBA5Ly64ZhqQBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"161576cd425b3efe3f132ab03bba6011b04edbee65e772a4deb8b9da45a9af89","last_reissued_at":"2026-07-05T01:59:07.771594Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:59:07.771594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Supercongruences involving Lucas sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2016-10-11T18:17:12Z","abstract_excerpt":"For $A,B\\in\\mathbb Z$, the Lucas sequence $u_n(A,B)\\ (n=0,1,2,\\ldots)$ are defined by $u_0(A,B)=0$, $u_1(A,B)=1$, and $u_{n+1}(A,B) = Au_n(A,B)-Bu_{n-1}(A,B)$ $(n=1,2,3,\\ldots).$ For any odd prime $p$ and positive integer $n$, we establish the new result $$\\frac{u_{pn}(A,B) - (\\frac{A^2-4B}p) u_n(A,B)}{pn} \\in \\mathbb Z_p,$$ where $(\\frac{\\cdot}p)$ is the Legendre symbol and $\\mathbb Z_p$ is the ring of $p$-adic integers.\n  Let $p$ be an odd prime and let $n$ be a positive integer. For any integer $m\\not\\equiv0\\pmod p$, we show that $$\\frac1{pn}\\bigg(\\sum_{k=0}^{pn-1} \\frac{\\binom{2k}k}{m^k} -"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03384","kind":"arxiv","version":8},"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/1610.03384/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":"1610.03384","created_at":"2026-07-05T01:59:07.771649+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.03384v8","created_at":"2026-07-05T01:59:07.771649+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.03384","created_at":"2026-07-05T01:59:07.771649+00:00"},{"alias_kind":"pith_short_12","alias_value":"CYKXNTKCLM7P","created_at":"2026-07-05T01:59:07.771649+00:00"},{"alias_kind":"pith_short_16","alias_value":"CYKXNTKCLM7P4PYT","created_at":"2026-07-05T01:59:07.771649+00:00"},{"alias_kind":"pith_short_8","alias_value":"CYKXNTKC","created_at":"2026-07-05T01:59:07.771649+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/CYKXNTKCLM7P4PYTFKYDXOTACG","json":"https://pith.science/pith/CYKXNTKCLM7P4PYTFKYDXOTACG.json","graph_json":"https://pith.science/api/pith-number/CYKXNTKCLM7P4PYTFKYDXOTACG/graph.json","events_json":"https://pith.science/api/pith-number/CYKXNTKCLM7P4PYTFKYDXOTACG/events.json","paper":"https://pith.science/paper/CYKXNTKC"},"agent_actions":{"view_html":"https://pith.science/pith/CYKXNTKCLM7P4PYTFKYDXOTACG","download_json":"https://pith.science/pith/CYKXNTKCLM7P4PYTFKYDXOTACG.json","view_paper":"https://pith.science/paper/CYKXNTKC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.03384&json=true","fetch_graph":"https://pith.science/api/pith-number/CYKXNTKCLM7P4PYTFKYDXOTACG/graph.json","fetch_events":"https://pith.science/api/pith-number/CYKXNTKCLM7P4PYTFKYDXOTACG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CYKXNTKCLM7P4PYTFKYDXOTACG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CYKXNTKCLM7P4PYTFKYDXOTACG/action/storage_attestation","attest_author":"https://pith.science/pith/CYKXNTKCLM7P4PYTFKYDXOTACG/action/author_attestation","sign_citation":"https://pith.science/pith/CYKXNTKCLM7P4PYTFKYDXOTACG/action/citation_signature","submit_replication":"https://pith.science/pith/CYKXNTKCLM7P4PYTFKYDXOTACG/action/replication_record"}},"created_at":"2026-07-05T01:59:07.771649+00:00","updated_at":"2026-07-05T01:59:07.771649+00:00"}