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The parametrizations are found for following Diophantine systems: \\begin{align*} (p^2\\pm q^2)^2-a^2 & =\\square_{1,2}\\,, \\\\[0.2cm] c^2-(p^2\\pm q^2)^2 & =\\square_{1,2}\\,, \\\\[0.2cm] a^2+(p^2\\pm q^2)^2 & =\\square_{1,2}\\,, \\\\[0.2cm] (p^2\\pm q^2)^2-a^2 & =(r^2\\pm s^2)^2. \\end{align*}"},"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":"1504.04584","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2015-04-16T09:18:06Z","cross_cats_sorted":[],"title_canon_sha256":"77eb383b8077dd256e9dc75783c70d1c7039016a1303aebbe6b4b98fca2f8c44","abstract_canon_sha256":"6fb4744d96419f61c3290edb16659a870057253f6e03acef345a8f9848fd6dd0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:32.013681Z","signature_b64":"xaKwpxE2Ya1jO2+UxTVE2cxXLafC0NeFq6767dLUowUthZl2Wm65DAYrgyu+P/WNzdYA9tC7BPuPRloQRZg6Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da5d7250fb950448e60ded55503cb3af31dfe10a7be4712bb6b16494f701b0e6","last_reissued_at":"2026-05-18T02:18:32.013074Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:32.013074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Diophantine Equations and Congruent Number Equation Solutions","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Mamuka Meskhishvili","submitted_at":"2015-04-16T09:18:06Z","abstract_excerpt":"By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\\;\\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and difference of the squares of the same rational numbers. The parametrizations are found for following Diophantine systems: \\begin{align*} (p^2\\pm q^2)^2-a^2 & =\\square_{1,2}\\,, \\\\[0.2cm] c^2-(p^2\\pm q^2)^2 & =\\square_{1,2}\\,, \\\\[0.2cm] a^2+(p^2\\pm q^2)^2 & =\\square_{1,2}\\,, \\\\[0.2cm] (p^2\\pm q^2)^2-a^2 & =(r^2\\pm s^2)^2. \\end{align*}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04584","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":"1504.04584","created_at":"2026-05-18T02:18:32.013167+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.04584v1","created_at":"2026-05-18T02:18:32.013167+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04584","created_at":"2026-05-18T02:18:32.013167+00:00"},{"alias_kind":"pith_short_12","alias_value":"3JOXEUH3SUCE","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"3JOXEUH3SUCERZQN","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"3JOXEUH3","created_at":"2026-05-18T12:29:02.477457+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/3JOXEUH3SUCERZQN5VKVAPFTV4","json":"https://pith.science/pith/3JOXEUH3SUCERZQN5VKVAPFTV4.json","graph_json":"https://pith.science/api/pith-number/3JOXEUH3SUCERZQN5VKVAPFTV4/graph.json","events_json":"https://pith.science/api/pith-number/3JOXEUH3SUCERZQN5VKVAPFTV4/events.json","paper":"https://pith.science/paper/3JOXEUH3"},"agent_actions":{"view_html":"https://pith.science/pith/3JOXEUH3SUCERZQN5VKVAPFTV4","download_json":"https://pith.science/pith/3JOXEUH3SUCERZQN5VKVAPFTV4.json","view_paper":"https://pith.science/paper/3JOXEUH3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.04584&json=true","fetch_graph":"https://pith.science/api/pith-number/3JOXEUH3SUCERZQN5VKVAPFTV4/graph.json","fetch_events":"https://pith.science/api/pith-number/3JOXEUH3SUCERZQN5VKVAPFTV4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3JOXEUH3SUCERZQN5VKVAPFTV4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3JOXEUH3SUCERZQN5VKVAPFTV4/action/storage_attestation","attest_author":"https://pith.science/pith/3JOXEUH3SUCERZQN5VKVAPFTV4/action/author_attestation","sign_citation":"https://pith.science/pith/3JOXEUH3SUCERZQN5VKVAPFTV4/action/citation_signature","submit_replication":"https://pith.science/pith/3JOXEUH3SUCERZQN5VKVAPFTV4/action/replication_record"}},"created_at":"2026-05-18T02:18:32.013167+00:00","updated_at":"2026-05-18T02:18:32.013167+00:00"}