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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","authors_text":"Long Li, Zhi-Hong Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-15T11:46:35Z","title":"Extensions of Stern's congruence for Euler numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3902","kind":"arxiv","version":1},"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:dd29042ed3f249b61aa40c0e2550e18511af17d3ff889096a7fd7e235100373c","target":"record","created_at":"2026-05-18T03:18:27Z","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":"487656c4a666fd4193a20e977982a40da6b02440983658fd76293f001f4202ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-15T11:46:35Z","title_canon_sha256":"e591e09ca9bc55696639dbd6b42149efad44df19cc3e5708adfc0e37ae9b9dd1"},"schema_version":"1.0","source":{"id":"1307.3902","kind":"arxiv","version":1}},"canonical_sha256":"0521e3b9da9c12fe32eb33e56ddd5e025c3cdcb805191ccec5e4301efe398ccb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0521e3b9da9c12fe32eb33e56ddd5e025c3cdcb805191ccec5e4301efe398ccb","first_computed_at":"2026-05-18T03:18:27.492502Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:27.492502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KEIiYGrHzCX9KLDD2kmJD2+BXAQF5rS7bpA+4Csc1jXBArM+HSVD+wopGT2OgFvEnGAMWFCN8XAaxDXwYe3lDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:27.493020Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.3902","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd29042ed3f249b61aa40c0e2550e18511af17d3ff889096a7fd7e235100373c","sha256:a303c25acd6df24cc279a5cf572ac9b3ddeb413c97527c005e16bdfdf1a5f987"],"state_sha256":"14df91f276e67e99d023520785d87ca8625a35e75e3cd8d647be8eef240cf1be"}