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Moreover, we prove that for any positive integer n and prime number p>2n+1 we have $$\\sum_{k=0}^{p-1+2n}E_kE_{p-1+2n-k}=s(n) (mod p)$$ where s(n) is an integer only depending on n."},"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":"1012.4774","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-21T19:33:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"df86853f3ca5ecbad783c3fc994ed56aefa4281bab6fb909ef8ad2ad9e05c218","abstract_canon_sha256":"8765ed96c38c53c7d437600656bcba32c71a08c26cfc968331c24f3bc30e9886"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:53.729056Z","signature_b64":"9lZzSN7Lo0Vo9M0og7vBo/BdYz7dMZGgnO1r/GpXApNaLqozh6wI59ZiFCRO+BKLwZJvgnc8R3JSCVgrNW1BBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b95fcfad6e396a50f2dae70c67abaa3d73a745dec6fa07eb3d850b1567fcc8ad","last_reissued_at":"2026-05-18T04:32:53.728676Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:53.728676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On convolutions of Euler numbers","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":"2010-12-21T19:33:07Z","abstract_excerpt":"We show that if p is an odd prime then $$\\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer n and prime number p>2n+1 we have $$\\sum_{k=0}^{p-1+2n}E_kE_{p-1+2n-k}=s(n) (mod p)$$ where s(n) is an integer only depending on n."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.4774","kind":"arxiv","version":1},"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":"1012.4774","created_at":"2026-05-18T04:32:53.728732+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.4774v1","created_at":"2026-05-18T04:32:53.728732+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.4774","created_at":"2026-05-18T04:32:53.728732+00:00"},{"alias_kind":"pith_short_12","alias_value":"XFP47LLOHFVF","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"XFP47LLOHFVFB4W2","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"XFP47LLO","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/XFP47LLOHFVFB4W244GGPK5KHV","json":"https://pith.science/pith/XFP47LLOHFVFB4W244GGPK5KHV.json","graph_json":"https://pith.science/api/pith-number/XFP47LLOHFVFB4W244GGPK5KHV/graph.json","events_json":"https://pith.science/api/pith-number/XFP47LLOHFVFB4W244GGPK5KHV/events.json","paper":"https://pith.science/paper/XFP47LLO"},"agent_actions":{"view_html":"https://pith.science/pith/XFP47LLOHFVFB4W244GGPK5KHV","download_json":"https://pith.science/pith/XFP47LLOHFVFB4W244GGPK5KHV.json","view_paper":"https://pith.science/paper/XFP47LLO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.4774&json=true","fetch_graph":"https://pith.science/api/pith-number/XFP47LLOHFVFB4W244GGPK5KHV/graph.json","fetch_events":"https://pith.science/api/pith-number/XFP47LLOHFVFB4W244GGPK5KHV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XFP47LLOHFVFB4W244GGPK5KHV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XFP47LLOHFVFB4W244GGPK5KHV/action/storage_attestation","attest_author":"https://pith.science/pith/XFP47LLOHFVFB4W244GGPK5KHV/action/author_attestation","sign_citation":"https://pith.science/pith/XFP47LLOHFVFB4W244GGPK5KHV/action/citation_signature","submit_replication":"https://pith.science/pith/XFP47LLOHFVFB4W244GGPK5KHV/action/replication_record"}},"created_at":"2026-05-18T04:32:53.728732+00:00","updated_at":"2026-05-18T04:32:53.728732+00:00"}