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For n=5, there is an integer-valued polynomial Pythagorean 5-tuple which parametrizes Pythagorean quintuples (similar to the case n=3). Pythagorean quadruples are closely related to (integer) Descartes quadruples (solutions of 2(b_1^2+b_2^2+b_3^2+b_4^2) = (b_1+b_2+b_3+b_4)^2), which we also parametrize by a Descartes quadruple of polynomials with integer coefficien"},"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":"1106.3493","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-06-17T14:15:03Z","cross_cats_sorted":[],"title_canon_sha256":"d46739ef0c0d0f280f95a1929c9cadd39ee8a43022700e064fe6623fa12cf7c8","abstract_canon_sha256":"89128e33ad24e0c4c38589ebb939f2b6278aa49786886b9e1d6a29e30202c3f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:27.441629Z","signature_b64":"8lFElQhF8ik3kYf/o0Izs/rW7zeNgOtWyUL8vi1XdbUIaF6hL3wEoBtavXcwFg1ZmDahoepCafHkDq43jOc9Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a785362687bb54ec1baf7b81bc465961729e02267bce87decc23979ca5ba098","last_reissued_at":"2026-05-18T04:05:27.441130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:27.441130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Leonid Vaserstein, Sophie Frisch","submitted_at":"2011-06-17T14:15:03Z","abstract_excerpt":"A Pythagorean n-tuple is an integer solution of x_1^2+...+x_{n-1}^2=x_n^2. 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Pythagorean quadruples are closely related to (integer) Descartes quadruples (solutions of 2(b_1^2+b_2^2+b_3^2+b_4^2) = (b_1+b_2+b_3+b_4)^2), which we also parametrize by a Descartes quadruple of polynomials with integer coefficien"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.3493","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":"1106.3493","created_at":"2026-05-18T04:05:27.441210+00:00"},{"alias_kind":"arxiv_version","alias_value":"1106.3493v2","created_at":"2026-05-18T04:05:27.441210+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.3493","created_at":"2026-05-18T04:05:27.441210+00:00"},{"alias_kind":"pith_short_12","alias_value":"RJ4FGYTIPO2U","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"RJ4FGYTIPO2U5QN2","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"RJ4FGYTI","created_at":"2026-05-18T12:26:41.206345+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/RJ4FGYTIPO2U5QN2664BXRDFSY","json":"https://pith.science/pith/RJ4FGYTIPO2U5QN2664BXRDFSY.json","graph_json":"https://pith.science/api/pith-number/RJ4FGYTIPO2U5QN2664BXRDFSY/graph.json","events_json":"https://pith.science/api/pith-number/RJ4FGYTIPO2U5QN2664BXRDFSY/events.json","paper":"https://pith.science/paper/RJ4FGYTI"},"agent_actions":{"view_html":"https://pith.science/pith/RJ4FGYTIPO2U5QN2664BXRDFSY","download_json":"https://pith.science/pith/RJ4FGYTIPO2U5QN2664BXRDFSY.json","view_paper":"https://pith.science/paper/RJ4FGYTI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1106.3493&json=true","fetch_graph":"https://pith.science/api/pith-number/RJ4FGYTIPO2U5QN2664BXRDFSY/graph.json","fetch_events":"https://pith.science/api/pith-number/RJ4FGYTIPO2U5QN2664BXRDFSY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RJ4FGYTIPO2U5QN2664BXRDFSY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RJ4FGYTIPO2U5QN2664BXRDFSY/action/storage_attestation","attest_author":"https://pith.science/pith/RJ4FGYTIPO2U5QN2664BXRDFSY/action/author_attestation","sign_citation":"https://pith.science/pith/RJ4FGYTIPO2U5QN2664BXRDFSY/action/citation_signature","submit_replication":"https://pith.science/pith/RJ4FGYTIPO2U5QN2664BXRDFSY/action/replication_record"}},"created_at":"2026-05-18T04:05:27.441210+00:00","updated_at":"2026-05-18T04:05:27.441210+00:00"}