{"id":"W4387772407","doi":"10.1007/978-3-031-37238-4_2","title":"Basic Theory of the Lucas Sequences","year":2023,"lang":"en","type":"book-chapter","venue":"CMS/CAIMS books in mathematics","topic":"Advanced Mathematical Theories and Applications","field":"Physics and Astronomy","cited_by":0,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Calgary","funders":"","keywords":"Mathematical proof; Lucas sequence; Mathematics; Lemma (botany); Rank (graph theory); Algebra over a field; Divisibility rule; Algebraic number; Lucas number; Calculus (dental); Fibonacci number; Pure mathematics; Discrete mathematics; Combinatorics; Fibonacci polynomials","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"],"consensus_categories":[],"category_scores_codex":[0.0004086641,0.0003623137,0.0006139504,0.00007371027,0.0001121796,0.00002733939,0.0006604317,0.000140702,0.0006856096],"category_scores_gemma":[0.00004372439,0.0002521725,0.0003176201,0.00008332446,0.0004978754,0.00004561476,0.0002784555,0.0004606475,0.0002007025],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0000352404,"about_ca_system_score_gemma":0.00008824017,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000002998649,"about_ca_topic_score_gemma":0.000002498445,"domain_scores_codex":[0.998359,0.00002238815,0.0007464986,0.0002950016,0.0003103967,0.0002666695],"domain_scores_gemma":[0.9974884,0.0008560067,0.0005595394,0.0009667053,0.00007320038,0.00005615545],"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":[0.000001977511,0.00004198249,0.000006766475,0.0001982543,0.00006044683,8.542691e-7,0.0004902352,0.00002793634,0.00003329491,0.9968045,0.0002913515,0.002042394],"study_design_scores_gemma":[0.0001141123,0.00001116574,0.000002868995,0.0007733766,0.00007876176,0.000001141728,0.0005227643,0.00009332663,0.0004008882,0.9897971,0.007951126,0.0002534183],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"other","genre_gemma":"other","genre_scores_codex":[0.0002069778,0.00006964554,0.0138896,0.00009582967,0.0001694835,0.0007283585,0.0001715661,0.00005817338,0.9846104],"genre_scores_gemma":[0.0142071,0.00003036084,0.005380018,0.00004094798,0.0002499926,0.0001534636,0.00002172507,0.0001664975,0.9797499],"genre_candidate":"other","genre_consensus":"other","teacher_disagreement_score":0.01400012,"threshold_uncertainty_score":0.999993,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03407109971941557,"score_gpt":0.2718469434585148,"score_spread":0.2377758437390992,"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."}}