{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:KPUQJY4DZIWPXSBUKFEMOPPNK7","short_pith_number":"pith:KPUQJY4D","schema_version":"1.0","canonical_sha256":"53e904e383ca2cfbc8345148c73ded57d5351dc72a7250e8e831f16dde2474c2","source":{"kind":"arxiv","id":"1908.02155","version":9},"attestation_state":"computed","paper":{"title":"Trigonometric identities and quadratic residues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.CA","authors_text":"Zhi-Wei Sun","submitted_at":"2019-08-06T15:59:18Z","abstract_excerpt":"In this paper we obtain some novel identities involving trigonometric functions. Let $n$ be any positive odd integer. We show that $$\\sum_{r=0}^{n-1}\\frac1{1+\\sin2\\pi\\frac{x+r}n+\\cos2\\pi\\frac{x+r}n} =\\frac{(-1)^{(n-1)/2}n}{1+(-1)^{(n-1)/2}\\sin 2\\pi x+\\cos 2\\pi x}$$ for any complex number with $x+1/2,x+(-1)^{(n-1)/2}/4\\not\\in\\mathbb Z$, and $$\\sum_{j,k=0}^{n-1}\\frac1{\\sin 2\\pi\\frac{x+j}n+\\sin2\\pi \\frac{y+k}n}=\\frac{(-1)^{(n-1)/2}n^2}{\\sin 2\\pi x+\\sin2\\pi y}$$ for all complex numbers $x$ and $y$ with $x+y,x-y-1/2\\not\\in\\mathbb Z$. We also determine the values of $\\prod_{k=1}^{(p-1)/2}(1+\\tan\\pi\\"},"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":"1908.02155","kind":"arxiv","version":9},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-06T15:59:18Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"09ee44eb5ee344bcac51c28515cb1c281338ac63fe28d3a148847d7cef3800a3","abstract_canon_sha256":"3890bf93af23684395e9d4971587152b9502c414910859b3064921ca9e090215"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:42:19.037837Z","signature_b64":"uJrJE4JZXXdxIgU6GDEaEwrJCqH3K35/vw0WgdiH1xxbZ4irM5Cgn53AaudF/Whp4dPYBRihd5Ex+LE80PLtDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53e904e383ca2cfbc8345148c73ded57d5351dc72a7250e8e831f16dde2474c2","last_reissued_at":"2026-07-05T03:42:19.037479Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:42:19.037479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Trigonometric identities and quadratic residues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.CA","authors_text":"Zhi-Wei Sun","submitted_at":"2019-08-06T15:59:18Z","abstract_excerpt":"In this paper we obtain some novel identities involving trigonometric functions. Let $n$ be any positive odd integer. We show that $$\\sum_{r=0}^{n-1}\\frac1{1+\\sin2\\pi\\frac{x+r}n+\\cos2\\pi\\frac{x+r}n} =\\frac{(-1)^{(n-1)/2}n}{1+(-1)^{(n-1)/2}\\sin 2\\pi x+\\cos 2\\pi x}$$ for any complex number with $x+1/2,x+(-1)^{(n-1)/2}/4\\not\\in\\mathbb Z$, and $$\\sum_{j,k=0}^{n-1}\\frac1{\\sin 2\\pi\\frac{x+j}n+\\sin2\\pi \\frac{y+k}n}=\\frac{(-1)^{(n-1)/2}n^2}{\\sin 2\\pi x+\\sin2\\pi y}$$ for all complex numbers $x$ and $y$ with $x+y,x-y-1/2\\not\\in\\mathbb Z$. We also determine the values of $\\prod_{k=1}^{(p-1)/2}(1+\\tan\\pi\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02155","kind":"arxiv","version":9},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.02155/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.02155","created_at":"2026-07-05T03:42:19.037535+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.02155v9","created_at":"2026-07-05T03:42:19.037535+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.02155","created_at":"2026-07-05T03:42:19.037535+00:00"},{"alias_kind":"pith_short_12","alias_value":"KPUQJY4DZIWP","created_at":"2026-07-05T03:42:19.037535+00:00"},{"alias_kind":"pith_short_16","alias_value":"KPUQJY4DZIWPXSBU","created_at":"2026-07-05T03:42:19.037535+00:00"},{"alias_kind":"pith_short_8","alias_value":"KPUQJY4D","created_at":"2026-07-05T03:42:19.037535+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/KPUQJY4DZIWPXSBUKFEMOPPNK7","json":"https://pith.science/pith/KPUQJY4DZIWPXSBUKFEMOPPNK7.json","graph_json":"https://pith.science/api/pith-number/KPUQJY4DZIWPXSBUKFEMOPPNK7/graph.json","events_json":"https://pith.science/api/pith-number/KPUQJY4DZIWPXSBUKFEMOPPNK7/events.json","paper":"https://pith.science/paper/KPUQJY4D"},"agent_actions":{"view_html":"https://pith.science/pith/KPUQJY4DZIWPXSBUKFEMOPPNK7","download_json":"https://pith.science/pith/KPUQJY4DZIWPXSBUKFEMOPPNK7.json","view_paper":"https://pith.science/paper/KPUQJY4D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.02155&json=true","fetch_graph":"https://pith.science/api/pith-number/KPUQJY4DZIWPXSBUKFEMOPPNK7/graph.json","fetch_events":"https://pith.science/api/pith-number/KPUQJY4DZIWPXSBUKFEMOPPNK7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KPUQJY4DZIWPXSBUKFEMOPPNK7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KPUQJY4DZIWPXSBUKFEMOPPNK7/action/storage_attestation","attest_author":"https://pith.science/pith/KPUQJY4DZIWPXSBUKFEMOPPNK7/action/author_attestation","sign_citation":"https://pith.science/pith/KPUQJY4DZIWPXSBUKFEMOPPNK7/action/citation_signature","submit_replication":"https://pith.science/pith/KPUQJY4DZIWPXSBUKFEMOPPNK7/action/replication_record"}},"created_at":"2026-07-05T03:42:19.037535+00:00","updated_at":"2026-07-05T03:42:19.037535+00:00"}