{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:AUQ6HOO2TQJP4MXLGPSW3XK6AJ","short_pith_number":"pith:AUQ6HOO2","schema_version":"1.0","canonical_sha256":"0521e3b9da9c12fe32eb33e56ddd5e025c3cdcb805191ccec5e4301efe398ccb","source":{"kind":"arxiv","id":"1307.3902","version":1},"attestation_state":"computed","paper":{"title":"Extensions of Stern's congruence for Euler numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Long Li, Zhi-Hong Sun","submitted_at":"2013-07-15T11:46:35Z","abstract_excerpt":"For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\\sum_{k=0}^{[n/2]}\\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$ can be viewed as a generalization of Euler numbers. Let $k$ and $m$ be positive integers, and let $b$ be a nonnegative integer. In this paper, we determine $E_{2^mk+b}^{(a)}$ modulo $ 2^{m+10}$ for $m\\ge 5$. For $m\\ge 5$ we also establish congruences for $U_{k\\varphi{(5^m)}+b},\\; E_{k\\varphi{(5^m)}+b},\\; S_{k\\varphi{(5^m)}+b}\\pmod{5^{m+5}}$ and $S_{k\\varphi"},"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":"1307.3902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-15T11:46:35Z","cross_cats_sorted":[],"title_canon_sha256":"e591e09ca9bc55696639dbd6b42149efad44df19cc3e5708adfc0e37ae9b9dd1","abstract_canon_sha256":"487656c4a666fd4193a20e977982a40da6b02440983658fd76293f001f4202ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:27.493020Z","signature_b64":"KEIiYGrHzCX9KLDD2kmJD2+BXAQF5rS7bpA+4Csc1jXBArM+HSVD+wopGT2OgFvEnGAMWFCN8XAaxDXwYe3lDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0521e3b9da9c12fe32eb33e56ddd5e025c3cdcb805191ccec5e4301efe398ccb","last_reissued_at":"2026-05-18T03:18:27.492502Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:27.492502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extensions of Stern's congruence for Euler numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Long Li, Zhi-Hong Sun","submitted_at":"2013-07-15T11:46:35Z","abstract_excerpt":"For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\\sum_{k=0}^{[n/2]}\\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$ can be viewed as a generalization of Euler numbers. Let $k$ and $m$ be positive integers, and let $b$ be a nonnegative integer. In this paper, we determine $E_{2^mk+b}^{(a)}$ modulo $ 2^{m+10}$ for $m\\ge 5$. For $m\\ge 5$ we also establish congruences for $U_{k\\varphi{(5^m)}+b},\\; E_{k\\varphi{(5^m)}+b},\\; S_{k\\varphi{(5^m)}+b}\\pmod{5^{m+5}}$ and $S_{k\\varphi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3902","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":"1307.3902","created_at":"2026-05-18T03:18:27.492599+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.3902v1","created_at":"2026-05-18T03:18:27.492599+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.3902","created_at":"2026-05-18T03:18:27.492599+00:00"},{"alias_kind":"pith_short_12","alias_value":"AUQ6HOO2TQJP","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"AUQ6HOO2TQJP4MXL","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"AUQ6HOO2","created_at":"2026-05-18T12:27:38.830355+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/AUQ6HOO2TQJP4MXLGPSW3XK6AJ","json":"https://pith.science/pith/AUQ6HOO2TQJP4MXLGPSW3XK6AJ.json","graph_json":"https://pith.science/api/pith-number/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/graph.json","events_json":"https://pith.science/api/pith-number/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/events.json","paper":"https://pith.science/paper/AUQ6HOO2"},"agent_actions":{"view_html":"https://pith.science/pith/AUQ6HOO2TQJP4MXLGPSW3XK6AJ","download_json":"https://pith.science/pith/AUQ6HOO2TQJP4MXLGPSW3XK6AJ.json","view_paper":"https://pith.science/paper/AUQ6HOO2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.3902&json=true","fetch_graph":"https://pith.science/api/pith-number/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/graph.json","fetch_events":"https://pith.science/api/pith-number/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/action/storage_attestation","attest_author":"https://pith.science/pith/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/action/author_attestation","sign_citation":"https://pith.science/pith/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/action/citation_signature","submit_replication":"https://pith.science/pith/AUQ6HOO2TQJP4MXLGPSW3XK6AJ/action/replication_record"}},"created_at":"2026-05-18T03:18:27.492599+00:00","updated_at":"2026-05-18T03:18:27.492599+00:00"}