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In this paper we mainly show that for $m,n,t\\in R_p$ with $m\\not\\e 0\\pmod p$, $$\\align &P_{[\\frac p6]}(t) \\e -\\Big(\\frac 3p\\Big)\\sum_{x=0}^{p-1}\\Big(\\frac{x^3-3x+2t}p\\Big)\\pmod p, &\\Big(\\sum_{x=0}^{p-1}\\Big(\\frac{x^3+mx+n}p\\Big)\\Big)^2\\equiv \\Big(\\frac{-3m}p\\Big) \\sum_{k=0}^{[p/6]}\\binom{2k}k\\binom{3k}k\\binom{6k}{3k} \\Big(\\frac{4m^3+27n^2}{12^3\\cdot 4m^3}\\Big)^k\\pmod p,$$ where $(\\frac ap)$ is the Legendre symbol and $[x]$ is the greatest integer"},"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":"1012.4234","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-20T03:16:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0f897bd19a33a02fb36a77835dd17dfc6ff57ac36e0bf193733edd61be697b3d","abstract_canon_sha256":"26205cba73338a93828cd8c8c4fc863caacce47d3e65738621e2ca7abc5e80e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:24.103331Z","signature_b64":"o66OTryyXkv1lucE4LBwH87U7nGUUsL2TLfCK4aix+NCRVY8B3ywh3wCVXExlgWuZOh2PHkb+WV7WNLHTOsYAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d43837feec2edb3c5a42017abab8305493062d69d3374ef08cb003dfc353836","last_reissued_at":"2026-05-18T03:42:24.102835Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:24.102835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences concerning Legendre polynomials III","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2010-12-20T03:16:57Z","abstract_excerpt":"Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is coprime to $p$. Let $\\{P_n(x)\\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\\in R_p$ with $m\\not\\e 0\\pmod p$, $$\\align &P_{[\\frac p6]}(t) \\e -\\Big(\\frac 3p\\Big)\\sum_{x=0}^{p-1}\\Big(\\frac{x^3-3x+2t}p\\Big)\\pmod p, &\\Big(\\sum_{x=0}^{p-1}\\Big(\\frac{x^3+mx+n}p\\Big)\\Big)^2\\equiv \\Big(\\frac{-3m}p\\Big) \\sum_{k=0}^{[p/6]}\\binom{2k}k\\binom{3k}k\\binom{6k}{3k} \\Big(\\frac{4m^3+27n^2}{12^3\\cdot 4m^3}\\Big)^k\\pmod p,$$ where $(\\frac ap)$ is the Legendre symbol and $[x]$ is the greatest integer"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.4234","kind":"arxiv","version":4},"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":"1012.4234","created_at":"2026-05-18T03:42:24.102917+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.4234v4","created_at":"2026-05-18T03:42:24.102917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.4234","created_at":"2026-05-18T03:42:24.102917+00:00"},{"alias_kind":"pith_short_12","alias_value":"BVBYG77OYLW3","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"BVBYG77OYLW3HRNE","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"BVBYG77O","created_at":"2026-05-18T12:26:05.355336+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/BVBYG77OYLW3HRNEEAL2XK4DAV","json":"https://pith.science/pith/BVBYG77OYLW3HRNEEAL2XK4DAV.json","graph_json":"https://pith.science/api/pith-number/BVBYG77OYLW3HRNEEAL2XK4DAV/graph.json","events_json":"https://pith.science/api/pith-number/BVBYG77OYLW3HRNEEAL2XK4DAV/events.json","paper":"https://pith.science/paper/BVBYG77O"},"agent_actions":{"view_html":"https://pith.science/pith/BVBYG77OYLW3HRNEEAL2XK4DAV","download_json":"https://pith.science/pith/BVBYG77OYLW3HRNEEAL2XK4DAV.json","view_paper":"https://pith.science/paper/BVBYG77O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.4234&json=true","fetch_graph":"https://pith.science/api/pith-number/BVBYG77OYLW3HRNEEAL2XK4DAV/graph.json","fetch_events":"https://pith.science/api/pith-number/BVBYG77OYLW3HRNEEAL2XK4DAV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BVBYG77OYLW3HRNEEAL2XK4DAV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BVBYG77OYLW3HRNEEAL2XK4DAV/action/storage_attestation","attest_author":"https://pith.science/pith/BVBYG77OYLW3HRNEEAL2XK4DAV/action/author_attestation","sign_citation":"https://pith.science/pith/BVBYG77OYLW3HRNEEAL2XK4DAV/action/citation_signature","submit_replication":"https://pith.science/pith/BVBYG77OYLW3HRNEEAL2XK4DAV/action/replication_record"}},"created_at":"2026-05-18T03:42:24.102917+00:00","updated_at":"2026-05-18T03:42:24.102917+00:00"}