{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:V7FYCAJS57HIXRIK3R43LIXDA6","short_pith_number":"pith:V7FYCAJS","schema_version":"1.0","canonical_sha256":"afcb810132efce8bc50adc79b5a2e307abd4ba993131f389147e3d900bfbe765","source":{"kind":"arxiv","id":"1012.3898","version":2},"attestation_state":"computed","paper":{"title":"Congruences concerning Legendre polynomials II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2010-12-17T15:00:29Z","abstract_excerpt":"Let $p>3$ be a prime, and let $m$ be an integer with $p\\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\\sum_{k=0}^{p-1}\\binom{2k}k^3/m^k\\pmod{p^2}$, $\\sum_{k=0}^{p-1}\\binom{2k}k\\b{4k}{2k}/m^k\\pmod p$ and $\\sum_{k=0}^{p-1}\\binom{2k}k^2\\b{4k}{2k}/m^k\\pmod {p^2}.$ In particular, we show that $\\sum_{k=0}^{\\frac{p-1}{2}}\\binom{2k}k^3\\equiv 0\\pmod {p^2}$ for $p\\equiv 3,5,6\\pmod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $ P_{[\\frac {p}{4}]}(t)\\equiv -\\big(\\frac{-6}{p}\\big)\\sum_{x=0}^{p-1} \\big(\\frac{x^3-3/2(3t+5)x-9t-7}{p}\\big)\\pmod p$ and d"},"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":"1012.3898","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T15:00:29Z","cross_cats_sorted":[],"title_canon_sha256":"7cd7df6853bf013b4c11aebec832c5d81aad77bbb88794b5256ffd7b092fb962","abstract_canon_sha256":"40a4ac93549d99ceb60336f72871f426989df7365894c19252c0f0df11d575d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:33.051361Z","signature_b64":"q57CrxCOMq2KDy3CGkGIRZUdXLzSgchOrpa/AcbsGPhfbSteB45Zw8ROto5tVp4tXs6ZOQOF5p2HJFxy7TX6Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afcb810132efce8bc50adc79b5a2e307abd4ba993131f389147e3d900bfbe765","last_reissued_at":"2026-05-18T03:49:33.050393Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:33.050393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences concerning Legendre polynomials II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2010-12-17T15:00:29Z","abstract_excerpt":"Let $p>3$ be a prime, and let $m$ be an integer with $p\\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\\sum_{k=0}^{p-1}\\binom{2k}k^3/m^k\\pmod{p^2}$, $\\sum_{k=0}^{p-1}\\binom{2k}k\\b{4k}{2k}/m^k\\pmod p$ and $\\sum_{k=0}^{p-1}\\binom{2k}k^2\\b{4k}{2k}/m^k\\pmod {p^2}.$ In particular, we show that $\\sum_{k=0}^{\\frac{p-1}{2}}\\binom{2k}k^3\\equiv 0\\pmod {p^2}$ for $p\\equiv 3,5,6\\pmod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $ P_{[\\frac {p}{4}]}(t)\\equiv -\\big(\\frac{-6}{p}\\big)\\sum_{x=0}^{p-1} \\big(\\frac{x^3-3/2(3t+5)x-9t-7}{p}\\big)\\pmod p$ and d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3898","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1012.3898","created_at":"2026-05-18T03:49:33.050559+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.3898v2","created_at":"2026-05-18T03:49:33.050559+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3898","created_at":"2026-05-18T03:49:33.050559+00:00"},{"alias_kind":"pith_short_12","alias_value":"V7FYCAJS57HI","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_16","alias_value":"V7FYCAJS57HIXRIK","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_8","alias_value":"V7FYCAJS","created_at":"2026-05-18T12:26:15.391820+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/V7FYCAJS57HIXRIK3R43LIXDA6","json":"https://pith.science/pith/V7FYCAJS57HIXRIK3R43LIXDA6.json","graph_json":"https://pith.science/api/pith-number/V7FYCAJS57HIXRIK3R43LIXDA6/graph.json","events_json":"https://pith.science/api/pith-number/V7FYCAJS57HIXRIK3R43LIXDA6/events.json","paper":"https://pith.science/paper/V7FYCAJS"},"agent_actions":{"view_html":"https://pith.science/pith/V7FYCAJS57HIXRIK3R43LIXDA6","download_json":"https://pith.science/pith/V7FYCAJS57HIXRIK3R43LIXDA6.json","view_paper":"https://pith.science/paper/V7FYCAJS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.3898&json=true","fetch_graph":"https://pith.science/api/pith-number/V7FYCAJS57HIXRIK3R43LIXDA6/graph.json","fetch_events":"https://pith.science/api/pith-number/V7FYCAJS57HIXRIK3R43LIXDA6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V7FYCAJS57HIXRIK3R43LIXDA6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V7FYCAJS57HIXRIK3R43LIXDA6/action/storage_attestation","attest_author":"https://pith.science/pith/V7FYCAJS57HIXRIK3R43LIXDA6/action/author_attestation","sign_citation":"https://pith.science/pith/V7FYCAJS57HIXRIK3R43LIXDA6/action/citation_signature","submit_replication":"https://pith.science/pith/V7FYCAJS57HIXRIK3R43LIXDA6/action/replication_record"}},"created_at":"2026-05-18T03:49:33.050559+00:00","updated_at":"2026-05-18T03:49:33.050559+00:00"}