{"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":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0004565458,0.0006029041,0.0004902885,0.0008390297,0.0005097662,0.0007312647,0.00102444,0.000502586,0.003934377],"category_scores_gemma":[0.001003131,0.0002224951,0.0007066931,0.0004752369,0.001721023,0.001792325,0.001089598,0.001688339,0.0008838779],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0006750337,"about_ca_system_score_gemma":0.0004376282,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0005945842,"about_ca_topic_score_gemma":0.001086014,"domain_scores_codex":[0.999854,0.0000420135,0.000007123923,0.00002405937,0.00005201534,0.00002086038],"domain_scores_gemma":[0.9998311,0.00004697048,0.00002838454,0.0000415646,0.00003283107,0.00001922238],"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":[0.00003244086,0.00002501663,0.0002280693,0.00005818859,0.000009963137,0.0001493285,0.0002788467,0.01610471,0.006282699,0.9642605,0.001156098,0.01141408],"study_design_scores_gemma":[0.00002813478,0.0000625944,0.0003751286,0.00002305369,0.00001508761,0.0002263218,0.0001184247,0.1131955,0.001911726,0.8730779,0.01094659,0.00001957712],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.2269085,0.001294037,0.6812485,0.002164146,0.0004576253,0.000129782,0.0001406817,0.0002154486,0.08744117],"genre_scores_gemma":[0.6618629,0.001779386,0.2977682,0.0004939876,0.0005129548,0.0001252088,0.0001496927,0.0001453743,0.0371624],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.003934377,"threshold_uncertainty_score":0.01316184,"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."}}