{"id":"W1977507299","doi":"10.4153/cmb-2002-045-2","title":"Criteria for Simultaneous Solutions of <i>X</i><sup>2</sup> − <i>DY</i><sup>2</sup> = <i>c</i> and <i>x</i><sup>2</sup> − <i>Dy</i><sup>2</sup> = −<i>c</i>","year":2002,"lang":"en","type":"article","venue":"Canadian Mathematical Bulletin","topic":"Advanced Differential Equations and Dynamical Systems","field":"Mathematics","cited_by":5,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Calgary","funders":"Natural Sciences and Engineering Research Council of Canada","keywords":"Mathematics; Diophantine equation; Pure mathematics; Square (algebra); Discrete mathematics; Algebra over a field; Geometry","routes":{"ca_aff":true,"ca_fund":true,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["metaepi_narrow","sts","research_integrity","insufficient_payload"],"consensus_categories":["metaepi_narrow","insufficient_payload"],"category_scores_codex":[0.002043792,0.00235579,0.003732582,0.0009276606,0.001804896,0.0006797129,0.001946086,0.001536422,0.009886362],"category_scores_gemma":[0.007337168,0.002323067,0.001361434,0.001409547,0.001517561,0.0005388577,0.0005961995,0.001555524,0.00278134],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0008619308,"about_ca_system_score_gemma":0.0005086958,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001593502,"about_ca_topic_score_gemma":0.0005285537,"domain_scores_codex":[0.9858897,0.0008426368,0.00446231,0.002641894,0.001852122,0.004311382],"domain_scores_gemma":[0.9832208,0.007915962,0.00105541,0.002945877,0.001173657,0.003688286],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"not_applicable","study_design_gemma":"simulation_or_modeling","study_design_scores_codex":[0.0008075165,0.006737003,0.00022582,0.0135222,0.002505053,0.0008223921,0.02430149,0.09045559,0.001259223,0.3997945,0.4449835,0.01458573],"study_design_scores_gemma":[0.005000305,0.0007144483,0.000002824759,0.00141597,0.001016602,0.0005263464,0.003405584,0.6996421,0.0001854025,0.09604931,0.1891304,0.002910803],"study_design_candidate":"simulation_or_modeling","study_design_consensus":null,"genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.4295164,0.007066762,0.4259285,0.02704431,0.001133653,0.02555878,0.019036,0.003397377,0.0613182],"genre_scores_gemma":[0.9381479,0.000338413,0.04150587,0.003426753,0.00139106,0.001466429,0.0005699524,0.0008746943,0.01227893],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.6091865,"threshold_uncertainty_score":0.9997598,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03571964801635275,"score_gpt":0.2664953362472899,"score_spread":0.2307756882309371,"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."}}