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For $n=0,1,2,\\ldots$ define $$d_n(x)=\\sum_{k=0}^n\\binom nk\\binom xk2^k$$ and $$s_n(x)=\\sum_{k=0}^n\\binom nk\\binom xk\\binom{x+k}k=\\sum_{k=0}^n\\binom nk(-1)^k\\binom xk\\binom{-1-x}k.$$ For any odd prime $p$ and $p$-adic integer $x$, we determine $\\sum_{k=0}^{p-1}(\\pm1)^kd_k(x)^2$ and $\\sum_{k=0}^{p-1}(2k+1)d_k(x)^2$ modulo $p^2$; for example, we establish the new $p$-adic congruence $$\\sum_{k=0}^{p-1}(-1)^kd_k(x)^2\\equiv(-1)^{\\langle x\\rangle_p}\\pmod{p^2},$$ where $\\langle x\\rangle_p$ denotes the least nonnegativ"},"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":"1512.00712","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-12-02T16:57:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6676c4df54c4525c665dd753746c78a1d4fdcf3a42bfda7fe1616bffb2287486","abstract_canon_sha256":"149eb754ef820b967cc158bd948834ee0f3441e767c18141afdc674d33ebde85"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:04.600723Z","signature_b64":"RbkwYlxriLWmkq03skti118PKSciKgXbkkKTjBrvkY2b0KCEXCGYzhJPxPD5hKQCFleuSMD37MGRn6gGwOCaBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"415164111ed13a7106772413d6c793ad78b9f4b32eb94e400150e48aac303d02","last_reissued_at":"2026-05-18T00:46:04.600281Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:04.600281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Supercongruences involving dual sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2015-12-02T16:57:59Z","abstract_excerpt":"In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\\ldots$ define $$d_n(x)=\\sum_{k=0}^n\\binom nk\\binom xk2^k$$ and $$s_n(x)=\\sum_{k=0}^n\\binom nk\\binom xk\\binom{x+k}k=\\sum_{k=0}^n\\binom nk(-1)^k\\binom xk\\binom{-1-x}k.$$ For any odd prime $p$ and $p$-adic integer $x$, we determine $\\sum_{k=0}^{p-1}(\\pm1)^kd_k(x)^2$ and $\\sum_{k=0}^{p-1}(2k+1)d_k(x)^2$ modulo $p^2$; for example, we establish the new $p$-adic congruence $$\\sum_{k=0}^{p-1}(-1)^kd_k(x)^2\\equiv(-1)^{\\langle x\\rangle_p}\\pmod{p^2},$$ where $\\langle x\\rangle_p$ denotes the least nonnegativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.00712","kind":"arxiv","version":5},"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":"1512.00712","created_at":"2026-05-18T00:46:04.600362+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.00712v5","created_at":"2026-05-18T00:46:04.600362+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.00712","created_at":"2026-05-18T00:46:04.600362+00:00"},{"alias_kind":"pith_short_12","alias_value":"IFIWIEI62E5H","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"IFIWIEI62E5HCBTX","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"IFIWIEI6","created_at":"2026-05-18T12:29:25.134429+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/IFIWIEI62E5HCBTXEQJ5NR4TVV","json":"https://pith.science/pith/IFIWIEI62E5HCBTXEQJ5NR4TVV.json","graph_json":"https://pith.science/api/pith-number/IFIWIEI62E5HCBTXEQJ5NR4TVV/graph.json","events_json":"https://pith.science/api/pith-number/IFIWIEI62E5HCBTXEQJ5NR4TVV/events.json","paper":"https://pith.science/paper/IFIWIEI6"},"agent_actions":{"view_html":"https://pith.science/pith/IFIWIEI62E5HCBTXEQJ5NR4TVV","download_json":"https://pith.science/pith/IFIWIEI62E5HCBTXEQJ5NR4TVV.json","view_paper":"https://pith.science/paper/IFIWIEI6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.00712&json=true","fetch_graph":"https://pith.science/api/pith-number/IFIWIEI62E5HCBTXEQJ5NR4TVV/graph.json","fetch_events":"https://pith.science/api/pith-number/IFIWIEI62E5HCBTXEQJ5NR4TVV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IFIWIEI62E5HCBTXEQJ5NR4TVV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IFIWIEI62E5HCBTXEQJ5NR4TVV/action/storage_attestation","attest_author":"https://pith.science/pith/IFIWIEI62E5HCBTXEQJ5NR4TVV/action/author_attestation","sign_citation":"https://pith.science/pith/IFIWIEI62E5HCBTXEQJ5NR4TVV/action/citation_signature","submit_replication":"https://pith.science/pith/IFIWIEI62E5HCBTXEQJ5NR4TVV/action/replication_record"}},"created_at":"2026-05-18T00:46:04.600362+00:00","updated_at":"2026-05-18T00:46:04.600362+00:00"}