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A positive integer $n$ is called a $(\\mathbb K,\\theta)$-congruent number if there is a triangle, called the $(\\mathbb K,\\theta, n)$-triangles, with sides in $\\mathbb K$ having $\\theta$ as an angle and $n\\alpha_\\theta$ as area, where ${\\alpha_\\theta}=\\sqrt{r^2-s^2}$. Consider the $(\\mathbb K,\\theta)$-congruent number elliptic curve $E_{n,\\theta}: y^2=x(x+(r+s)n)(x-(r-s)n)$ de"},"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":"1412.3258","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.NT","submitted_at":"2014-12-10T11:10:21Z","cross_cats_sorted":[],"title_canon_sha256":"2041903c92232b7626acd6ec23997df499d6d0a634ba8488d2dcd84f95bba55a","abstract_canon_sha256":"c5899682197c6ee241b588454aa3b447048819952055647e6f1523b9cf6329f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:31:29.649127Z","signature_b64":"+mvyJI3vJhBO0EzlLVJhHZ6afrjhybf/eA6ieB5BggIKUPPwqTJqa372FGgy2d6NDf+qEmHucdG3qbebQF7YCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1146766c22512e9e90a0a7d3e13f0e2aaaa3888332993b5bb136f0d411e7439f","last_reissued_at":"2026-05-18T02:31:29.648715Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:31:29.648715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On $\\theta$-congruent numbers on real quadratic number fields","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ali S. Janfada, Sajad Salami","submitted_at":"2014-12-10T11:10:21Z","abstract_excerpt":"Let ${\\mathbb K}={\\mathbb Q}(\\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0 < \\theta< \\pi $ has rational cosine, say $\\cos (\\theta)=s/r$ with $0< |s|<r$ and $\\gcd(r,s)=1$. A positive integer $n$ is called a $(\\mathbb K,\\theta)$-congruent number if there is a triangle, called the $(\\mathbb K,\\theta, n)$-triangles, with sides in $\\mathbb K$ having $\\theta$ as an angle and $n\\alpha_\\theta$ as area, where ${\\alpha_\\theta}=\\sqrt{r^2-s^2}$. Consider the $(\\mathbb K,\\theta)$-congruent number elliptic curve $E_{n,\\theta}: y^2=x(x+(r+s)n)(x-(r-s)n)$ de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.3258","kind":"arxiv","version":2},"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":"1412.3258","created_at":"2026-05-18T02:31:29.648782+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.3258v2","created_at":"2026-05-18T02:31:29.648782+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.3258","created_at":"2026-05-18T02:31:29.648782+00:00"},{"alias_kind":"pith_short_12","alias_value":"CFDHM3BCKEXJ","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"CFDHM3BCKEXJ5EFA","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"CFDHM3BC","created_at":"2026-05-18T12:28:22.404517+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/CFDHM3BCKEXJ5EFAU7J6CPYOFK","json":"https://pith.science/pith/CFDHM3BCKEXJ5EFAU7J6CPYOFK.json","graph_json":"https://pith.science/api/pith-number/CFDHM3BCKEXJ5EFAU7J6CPYOFK/graph.json","events_json":"https://pith.science/api/pith-number/CFDHM3BCKEXJ5EFAU7J6CPYOFK/events.json","paper":"https://pith.science/paper/CFDHM3BC"},"agent_actions":{"view_html":"https://pith.science/pith/CFDHM3BCKEXJ5EFAU7J6CPYOFK","download_json":"https://pith.science/pith/CFDHM3BCKEXJ5EFAU7J6CPYOFK.json","view_paper":"https://pith.science/paper/CFDHM3BC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.3258&json=true","fetch_graph":"https://pith.science/api/pith-number/CFDHM3BCKEXJ5EFAU7J6CPYOFK/graph.json","fetch_events":"https://pith.science/api/pith-number/CFDHM3BCKEXJ5EFAU7J6CPYOFK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CFDHM3BCKEXJ5EFAU7J6CPYOFK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CFDHM3BCKEXJ5EFAU7J6CPYOFK/action/storage_attestation","attest_author":"https://pith.science/pith/CFDHM3BCKEXJ5EFAU7J6CPYOFK/action/author_attestation","sign_citation":"https://pith.science/pith/CFDHM3BCKEXJ5EFAU7J6CPYOFK/action/citation_signature","submit_replication":"https://pith.science/pith/CFDHM3BCKEXJ5EFAU7J6CPYOFK/action/replication_record"}},"created_at":"2026-05-18T02:31:29.648782+00:00","updated_at":"2026-05-18T02:31:29.648782+00:00"}