{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:BXH6LALRHAENLS2RP4CI2LK2VR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c309637381f2f1cbc416796532e1f709cc60faaa704548a355de32cf7c655e54","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-30T11:30:57Z","title_canon_sha256":"2073f3c552fde342f3321b6db39bf39d1b0638fb386ad06625c2fdd5cbd39b76"},"schema_version":"1.0","source":{"id":"1605.09179","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.09179","created_at":"2026-05-18T01:13:21Z"},{"alias_kind":"arxiv_version","alias_value":"1605.09179v1","created_at":"2026-05-18T01:13:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.09179","created_at":"2026-05-18T01:13:21Z"},{"alias_kind":"pith_short_12","alias_value":"BXH6LALRHAEN","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"BXH6LALRHAENLS2R","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"BXH6LALR","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:19c21b8c7d967bfa3111b5235ae284ca2c24c43595527cb7c6f9aa51ccf71c98","target":"graph","created_at":"2026-05-18T01:13:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $p>3$ be a prime, and let $a$ be a rational p-adic integer with $a\\not\\equiv 0\\pmod p$. In this paper we establish congruences for $$\\sum_{k=1}^{(p-1)/2}\\frac{\\binom ak\\binom{-1-a}k}k, \\quad\\sum_{k=0}^{(p-1)/2}k\\binom ak\\binom{-1-a}k \\quad\\text{and}\\quad\\sum_{k=0}^{(p-1)/2}\\frac{\\binom ak\\binom{-1-a}k}{2k-1}\\pmod {p^2}$$ in terms of Bernoulli and Euler polynomials. We also give some transformation formulas for congruences modulo $p^2$.","authors_text":"Zhi-Hong Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-30T11:30:57Z","title":"New super congruences involving Bernoulli and Euler polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.09179","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":"sha256:02fb87119ed0aa192ca8b99085f45499a2e0c3cc2bfae2f9e718636918185ec1","target":"record","created_at":"2026-05-18T01:13:21Z","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":"c309637381f2f1cbc416796532e1f709cc60faaa704548a355de32cf7c655e54","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-30T11:30:57Z","title_canon_sha256":"2073f3c552fde342f3321b6db39bf39d1b0638fb386ad06625c2fdd5cbd39b76"},"schema_version":"1.0","source":{"id":"1605.09179","kind":"arxiv","version":1}},"canonical_sha256":"0dcfe581713808d5cb517f048d2d5aac766fad6192b6c534201434bd87af57ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0dcfe581713808d5cb517f048d2d5aac766fad6192b6c534201434bd87af57ab","first_computed_at":"2026-05-18T01:13:21.852372Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:21.852372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5yhmWTyheZLMCkTpyK5Ux6QPwNEneB3t51pD8vOjyA/PLcM7h3jFCmUtyH/Aq9cMIfGZrih0wjmpuNtrN2U5DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:21.852927Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.09179","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02fb87119ed0aa192ca8b99085f45499a2e0c3cc2bfae2f9e718636918185ec1","sha256:19c21b8c7d967bfa3111b5235ae284ca2c24c43595527cb7c6f9aa51ccf71c98"],"state_sha256":"51a32ec39ded2a39be0031ab252d3a3b552bceb41857cbcb95354d7cf3b0a1b9"}