{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:YYSL5DDGJJV33QEHFB7TZKG7YG","short_pith_number":"pith:YYSL5DDG","schema_version":"1.0","canonical_sha256":"c624be8c664a6bbdc087287f3ca8dfc1b9e3e919e3855608817c3fb0b93d600f","source":{"kind":"arxiv","id":"1011.3487","version":8},"attestation_state":"computed","paper":{"title":"Supercongruences motivated by e","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":"2010-11-15T19:59:35Z","abstract_excerpt":"In this paper we establish some new supercongruences motivated by the well-known fact $\\lim_{n\\to\\infty}(1+1/n)^n=e$. Let $p>3$ be a prime. We prove that $$\\sum_{k=0}^{p-1}\\binom{-1/(p+1)}k^{p+1}\\equiv 0\\ \\pmod{p^5}\\ \\ \\ \\mbox{and}\\ \\ \\ \\sum_{k=0}^{p-1}\\binom{1/(p-1)}k^{p-1}\\equiv \\frac{2}{3}p^4B_{p-3}\\ \\pmod{p^5},$$ where $B_0,B_1,B_2,\\ldots$ are Bernoulli numbers. We also show that for any $a\\in\\mathbb Z$ with $p\\nmid a$ we have $$\\sum_{k=1}^{p-1}\\frac1k\\left(1+\\frac ak\\right)^k\\equiv -1\\pmod{p}\\ \\ \\ \\mbox{and}\\ \\ \\ \\sum_{k=1}^{p-1}\\frac1{k^2}\\left(1+\\frac ak\\right)^k\\equiv 1+\\frac 1{2a}\\pmo"},"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":"1011.3487","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-11-15T19:59:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"37492360756a04ef5680e044d6e607eb1fbecf7919a98958521c3e3aa7f7363a","abstract_canon_sha256":"cc35221419b7352ac181ce79d82088de2ea5e36adcb035bceb3d1c6f50ffccff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:11.105761Z","signature_b64":"XB5R60qz64ZTY52FJ15J2dVOpMzPQYqztUE3YvTen0LR9BSB5ICV1DcW0GZKceGfENNlK1aJXJaq8Nynl38EAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c624be8c664a6bbdc087287f3ca8dfc1b9e3e919e3855608817c3fb0b93d600f","last_reissued_at":"2026-05-18T02:26:11.105102Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:11.105102Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Supercongruences motivated by e","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":"2010-11-15T19:59:35Z","abstract_excerpt":"In this paper we establish some new supercongruences motivated by the well-known fact $\\lim_{n\\to\\infty}(1+1/n)^n=e$. Let $p>3$ be a prime. We prove that $$\\sum_{k=0}^{p-1}\\binom{-1/(p+1)}k^{p+1}\\equiv 0\\ \\pmod{p^5}\\ \\ \\ \\mbox{and}\\ \\ \\ \\sum_{k=0}^{p-1}\\binom{1/(p-1)}k^{p-1}\\equiv \\frac{2}{3}p^4B_{p-3}\\ \\pmod{p^5},$$ where $B_0,B_1,B_2,\\ldots$ are Bernoulli numbers. We also show that for any $a\\in\\mathbb Z$ with $p\\nmid a$ we have $$\\sum_{k=1}^{p-1}\\frac1k\\left(1+\\frac ak\\right)^k\\equiv -1\\pmod{p}\\ \\ \\ \\mbox{and}\\ \\ \\ \\sum_{k=1}^{p-1}\\frac1{k^2}\\left(1+\\frac ak\\right)^k\\equiv 1+\\frac 1{2a}\\pmo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.3487","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":""},"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":"1011.3487","created_at":"2026-05-18T02:26:11.105194+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.3487v8","created_at":"2026-05-18T02:26:11.105194+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.3487","created_at":"2026-05-18T02:26:11.105194+00:00"},{"alias_kind":"pith_short_12","alias_value":"YYSL5DDGJJV3","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"YYSL5DDGJJV33QEH","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"YYSL5DDG","created_at":"2026-05-18T12:26:17.028572+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/YYSL5DDGJJV33QEHFB7TZKG7YG","json":"https://pith.science/pith/YYSL5DDGJJV33QEHFB7TZKG7YG.json","graph_json":"https://pith.science/api/pith-number/YYSL5DDGJJV33QEHFB7TZKG7YG/graph.json","events_json":"https://pith.science/api/pith-number/YYSL5DDGJJV33QEHFB7TZKG7YG/events.json","paper":"https://pith.science/paper/YYSL5DDG"},"agent_actions":{"view_html":"https://pith.science/pith/YYSL5DDGJJV33QEHFB7TZKG7YG","download_json":"https://pith.science/pith/YYSL5DDGJJV33QEHFB7TZKG7YG.json","view_paper":"https://pith.science/paper/YYSL5DDG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.3487&json=true","fetch_graph":"https://pith.science/api/pith-number/YYSL5DDGJJV33QEHFB7TZKG7YG/graph.json","fetch_events":"https://pith.science/api/pith-number/YYSL5DDGJJV33QEHFB7TZKG7YG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YYSL5DDGJJV33QEHFB7TZKG7YG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YYSL5DDGJJV33QEHFB7TZKG7YG/action/storage_attestation","attest_author":"https://pith.science/pith/YYSL5DDGJJV33QEHFB7TZKG7YG/action/author_attestation","sign_citation":"https://pith.science/pith/YYSL5DDGJJV33QEHFB7TZKG7YG/action/citation_signature","submit_replication":"https://pith.science/pith/YYSL5DDGJJV33QEHFB7TZKG7YG/action/replication_record"}},"created_at":"2026-05-18T02:26:11.105194+00:00","updated_at":"2026-05-18T02:26:11.105194+00:00"}