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Sun: \\sum_{k=1}^{p-1}\\frac{D_k}{k} \\equiv -\\frac{2^{p-1}-1}{p} \\pmod p,\\\\ \\sum_{k=1}^{p-1}\\frac{H_k}{k 2^k}\\equiv 0 \\pmod{p},\\quad p\\geqslant 5, where p is a prime, D_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k} are the Delannoy numbers, and H_n=\\sum_{k=1}^n\\frac{1}{k} are the harmonic numbers. We also prove that, for any positive integer m and prime p>m+1, \\sum_{1\\leqslant k_1\\leqslant \\cdots \\leqslant k_m\\leqslant p-1}\\frac{1}{k_1\\cdots k_m 2^{k_m}} \\equiv\\frac{1}{2}\\sum_{k=1}^{p-1}\\frac{(-1)^{k-1}}{k^m} \\pmod p, which is a multiple g"},"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":"1312.6541","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-23T12:43:00Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ee3de1e603eeb70baf09c5f933cb533fa754e77a2a24a5ffb78ad6520fe57ee8","abstract_canon_sha256":"79ea8d8b8e3b9d950c53e1cede3d9bef5f720181c40f6eacb0444d159da73c12"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:57.981241Z","signature_b64":"6UxKqQBsPuOg5M+z6x2O4q9sb2NPdov+d/EqCtvZf/A3E/Rh4q00D2ZCyQVJ/neZff+JSorqd4Gacr87B7F5Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e960e9c2dcc5d93482b8f819839de8834ff3a2b9609560d6fa8efe0fbdf5fb77","last_reissued_at":"2026-05-18T03:03:57.980479Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:57.980479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some congruences related to the q-Fermat quotients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Victor J. W. Guo","submitted_at":"2013-12-23T12:43:00Z","abstract_excerpt":"We give q-analogues of the following congruences by Z.-W. Sun: \\sum_{k=1}^{p-1}\\frac{D_k}{k} \\equiv -\\frac{2^{p-1}-1}{p} \\pmod p,\\\\ \\sum_{k=1}^{p-1}\\frac{H_k}{k 2^k}\\equiv 0 \\pmod{p},\\quad p\\geqslant 5, where p is a prime, D_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k} are the Delannoy numbers, and H_n=\\sum_{k=1}^n\\frac{1}{k} are the harmonic numbers. We also prove that, for any positive integer m and prime p>m+1, \\sum_{1\\leqslant k_1\\leqslant \\cdots \\leqslant k_m\\leqslant p-1}\\frac{1}{k_1\\cdots k_m 2^{k_m}} \\equiv\\frac{1}{2}\\sum_{k=1}^{p-1}\\frac{(-1)^{k-1}}{k^m} \\pmod p, which is a multiple g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6541","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":"1312.6541","created_at":"2026-05-18T03:03:57.980610+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.6541v1","created_at":"2026-05-18T03:03:57.980610+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.6541","created_at":"2026-05-18T03:03:57.980610+00:00"},{"alias_kind":"pith_short_12","alias_value":"5FQOTQW4YXMT","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"5FQOTQW4YXMTJAVY","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"5FQOTQW4","created_at":"2026-05-18T12:27:34.582898+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/5FQOTQW4YXMTJAVY7AMYHHPIQN","json":"https://pith.science/pith/5FQOTQW4YXMTJAVY7AMYHHPIQN.json","graph_json":"https://pith.science/api/pith-number/5FQOTQW4YXMTJAVY7AMYHHPIQN/graph.json","events_json":"https://pith.science/api/pith-number/5FQOTQW4YXMTJAVY7AMYHHPIQN/events.json","paper":"https://pith.science/paper/5FQOTQW4"},"agent_actions":{"view_html":"https://pith.science/pith/5FQOTQW4YXMTJAVY7AMYHHPIQN","download_json":"https://pith.science/pith/5FQOTQW4YXMTJAVY7AMYHHPIQN.json","view_paper":"https://pith.science/paper/5FQOTQW4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.6541&json=true","fetch_graph":"https://pith.science/api/pith-number/5FQOTQW4YXMTJAVY7AMYHHPIQN/graph.json","fetch_events":"https://pith.science/api/pith-number/5FQOTQW4YXMTJAVY7AMYHHPIQN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5FQOTQW4YXMTJAVY7AMYHHPIQN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5FQOTQW4YXMTJAVY7AMYHHPIQN/action/storage_attestation","attest_author":"https://pith.science/pith/5FQOTQW4YXMTJAVY7AMYHHPIQN/action/author_attestation","sign_citation":"https://pith.science/pith/5FQOTQW4YXMTJAVY7AMYHHPIQN/action/citation_signature","submit_replication":"https://pith.science/pith/5FQOTQW4YXMTJAVY7AMYHHPIQN/action/replication_record"}},"created_at":"2026-05-18T03:03:57.980610+00:00","updated_at":"2026-05-18T03:03:57.980610+00:00"}