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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","authors_text":"Zhi-Hong Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T15:00:29Z","title":"Congruences concerning Legendre polynomials II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3898","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:38d6f1b3c7401badf3cf57005faef9f483247eb4c1595956550d710b78a0ddd2","target":"record","created_at":"2026-05-18T03:49:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"40a4ac93549d99ceb60336f72871f426989df7365894c19252c0f0df11d575d3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T15:00:29Z","title_canon_sha256":"7cd7df6853bf013b4c11aebec832c5d81aad77bbb88794b5256ffd7b092fb962"},"schema_version":"1.0","source":{"id":"1012.3898","kind":"arxiv","version":2}},"canonical_sha256":"afcb810132efce8bc50adc79b5a2e307abd4ba993131f389147e3d900bfbe765","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"afcb810132efce8bc50adc79b5a2e307abd4ba993131f389147e3d900bfbe765","first_computed_at":"2026-05-18T03:49:33.050393Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:33.050393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q57CrxCOMq2KDy3CGkGIRZUdXLzSgchOrpa/AcbsGPhfbSteB45Zw8ROto5tVp4tXs6ZOQOF5p2HJFxy7TX6Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:33.051361Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.3898","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:38d6f1b3c7401badf3cf57005faef9f483247eb4c1595956550d710b78a0ddd2","sha256:b5ad872200808a57c55c32ba438506c720a4c91a1b962486c2ae75f504cef13b"],"state_sha256":"9e760f8ed3af7f463b3bbdc163e86d49d5375ea4afb6fa7dcc8d411bd5dd9302"}