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We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$."},"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":"1004.2193","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-04-13T14:03:27Z","cross_cats_sorted":[],"title_canon_sha256":"4272dddb32df659d100705e37bf3f03c190cb51ad5d74ea74f3c63c874c51de8","abstract_canon_sha256":"d03ffd68c29fc789b9dd43e906fe2cee7a70be1225468221281ab1066e847c8e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:20.305608Z","signature_b64":"p0ZLl8WpRmHlc+XW7kwSaoSTjZi4JM1F5Kh2FrQf+1BQFJy9tKPJlCNhzG7HQrmOaJ8TDE8kO/LT2nHX9L2UAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"169d3634eae5a4daf8cc529a1fd52a64e02c11e6ebfe932918b591891762c1b8","last_reissued_at":"2026-05-18T04:40:20.304812Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:20.304812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the simplest sextic fields and related Thue equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Akinari Hoshi","submitted_at":"2010-04-13T14:03:27Z","abstract_excerpt":"We consider the parametric family of sextic Thue equations \\[ x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\\lambda \\] where $m\\in\\mathbb{Z}$ is an integer and $\\lambda$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.2193","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":"1004.2193","created_at":"2026-05-18T04:40:20.304936+00:00"},{"alias_kind":"arxiv_version","alias_value":"1004.2193v2","created_at":"2026-05-18T04:40:20.304936+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.2193","created_at":"2026-05-18T04:40:20.304936+00:00"},{"alias_kind":"pith_short_12","alias_value":"C2OTMNHK4WSN","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"C2OTMNHK4WSNV6GM","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"C2OTMNHK","created_at":"2026-05-18T12:26:05.355336+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/C2OTMNHK4WSNV6GMKKNB7VJKMT","json":"https://pith.science/pith/C2OTMNHK4WSNV6GMKKNB7VJKMT.json","graph_json":"https://pith.science/api/pith-number/C2OTMNHK4WSNV6GMKKNB7VJKMT/graph.json","events_json":"https://pith.science/api/pith-number/C2OTMNHK4WSNV6GMKKNB7VJKMT/events.json","paper":"https://pith.science/paper/C2OTMNHK"},"agent_actions":{"view_html":"https://pith.science/pith/C2OTMNHK4WSNV6GMKKNB7VJKMT","download_json":"https://pith.science/pith/C2OTMNHK4WSNV6GMKKNB7VJKMT.json","view_paper":"https://pith.science/paper/C2OTMNHK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1004.2193&json=true","fetch_graph":"https://pith.science/api/pith-number/C2OTMNHK4WSNV6GMKKNB7VJKMT/graph.json","fetch_events":"https://pith.science/api/pith-number/C2OTMNHK4WSNV6GMKKNB7VJKMT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C2OTMNHK4WSNV6GMKKNB7VJKMT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C2OTMNHK4WSNV6GMKKNB7VJKMT/action/storage_attestation","attest_author":"https://pith.science/pith/C2OTMNHK4WSNV6GMKKNB7VJKMT/action/author_attestation","sign_citation":"https://pith.science/pith/C2OTMNHK4WSNV6GMKKNB7VJKMT/action/citation_signature","submit_replication":"https://pith.science/pith/C2OTMNHK4WSNV6GMKKNB7VJKMT/action/replication_record"}},"created_at":"2026-05-18T04:40:20.304936+00:00","updated_at":"2026-05-18T04:40:20.304936+00:00"}