{"id":"W2964118337","doi":"10.1142/s1793042111004472","title":"A REFINED MODULAR APPROACH TO THE DIOPHANTINE EQUATION x<sup>2</sup> + y<sup>2n</sup> = z<sup>3</sup>","year":2011,"lang":"en","type":"article","venue":"International Journal of Number Theory","topic":"Algebraic Geometry and Number Theory","field":"Mathematics","cited_by":12,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of British Columbia","funders":"","keywords":"Diophantine equation; Integer (computer science); Mathematics; Coprime integers; Fermat's Last Theorem; Modular form; Number theory; Diophantine set; Galois module; Prime (order theory); Type (biology); Discrete mathematics; Combinatorics; Pure mathematics","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["metaepi_narrow","insufficient_payload"],"consensus_categories":["insufficient_payload"],"category_scores_codex":[0.005748754,0.0006674914,0.0008754572,0.0005856098,0.0003259823,0.0002158062,0.002594508,0.0003446,0.005841352],"category_scores_gemma":[0.003248465,0.0005103017,0.0007469196,0.0006418515,0.0002893592,0.0009202261,0.0005020394,0.001143758,0.0009548162],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0002930651,"about_ca_system_score_gemma":0.0001977732,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00002910386,"about_ca_topic_score_gemma":0.000001332222,"domain_scores_codex":[0.9934939,0.001180265,0.001885292,0.0006267142,0.002065231,0.0007486691],"domain_scores_gemma":[0.9942762,0.001846008,0.001150687,0.0009566107,0.001359948,0.0004105056],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.004988009,0.00366142,0.001267058,0.0001897049,0.004409885,0.0004305009,0.05064741,0.02198443,0.0002665123,0.8640294,0.03157251,0.01655317],"study_design_scores_gemma":[0.004001318,0.0003789406,0.0006993224,0.0006176856,0.0005770946,0.003008944,0.01012579,0.02872437,0.001282111,0.9225593,0.02689948,0.001125596],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.8207733,0.0002668429,0.1232606,0.001923071,0.0008277805,0.0007926552,0.0002255068,0.0001714291,0.05175886],"genre_scores_gemma":[0.9751461,0.00004912211,0.01495419,0.001521483,0.002403307,0.00006562758,0.00007711589,0.00015137,0.00563164],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.1543728,"threshold_uncertainty_score":0.999823,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.04951312201273072,"score_gpt":0.2864905686183558,"score_spread":0.2369774466056251,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}