{"paper":{"title":"On the shortest open cubic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The equation 7x³ + 2y³ = 3z² + 1 has no integer solutions.","cross_cats":[],"primary_cat":"math.GM","authors_text":"Ashleigh Ratcliffe, Bogdan Grechuk","submitted_at":"2026-03-31T14:52:42Z","abstract_excerpt":"We use cubic reciprocity to prove that the equation $7x^3+2y^3=3z^2+1$ has no integer solutions. Prior to this work, it was the shortest cubic equation for which the existence of integer solutions remained open. We conclude with a list of the new shortest open cubic equations."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We use cubic reciprocity to prove that the equation 7x^3+2y^3=3z^2+1 has no integer solutions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Cubic reciprocity can be applied directly to this equation without additional unstated conditions or case distinctions that might allow solutions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves no integer solutions for 7x^3 + 2y^3 = 3z^2 + 1 and updates the list of shortest open cubic equations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The equation 7x³ + 2y³ = 3z² + 1 has no integer solutions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5be0777c306774afe417c9fd320723ab3cc2cdb77aef0e5bc04f20f6929143ae"},"source":{"id":"2603.29831","kind":"arxiv","version":2},"verdict":{"id":"f47d22bf-c041-4c86-b98b-1412380f51a4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:04:06.602965Z","strongest_claim":"We use cubic reciprocity to prove that the equation 7x^3+2y^3=3z^2+1 has no integer solutions.","one_line_summary":"Proves no integer solutions for 7x^3 + 2y^3 = 3z^2 + 1 and updates the list of shortest open cubic equations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Cubic reciprocity can be applied directly to this equation without additional unstated conditions or case distinctions that might allow solutions.","pith_extraction_headline":"The equation 7x³ + 2y³ = 3z² + 1 has no integer solutions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.29831/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":16,"sample":[{"doi":"","year":2025,"title":"Brauer groups of certain affine cubic surfaces.arXiv preprint arXiv:2509.16042, 2025","work_id":"29f2fdfa-87b1-4755-839d-3a858aaf2c10","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1961,"title":"The decision problem for exponential Diophantine equations.Ann","work_id":"871f0974-66a7-4e27-945d-08f6c585f301","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"A systematic approach to diophantine equations: open problems.arXiv preprint arXiv:2404.08518, 2024","work_id":"bea2eb24-8e4a-449b-81f7-ab59e412fccc","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Springer, Cham, [2024] ©2024","work_id":"86c7d00e-2a31-4815-b5fb-1ec7235bee7f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Cubic diophantine equations: integer solutions beyond direct search.The American Mathematical Monthly, to appear, 2026","work_id":"c4ce60c3-9e70-40ca-8b7d-873417cdb826","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":16,"snapshot_sha256":"8d6b9f16075f3fd6c7e3da58b07d69c35be849fecbb1d8dab7b8e5f3ec3aba11","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"448e4934a121beead785a37e5395101c693d8c998120b71073f769494dc1a77a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}