{"id":"W2156136629","doi":"10.1142/s021949880500123x","title":"PYTHAGOREAN TRIPLES AND UNITS IN INTEGRAL GROUP RINGS","year":2005,"lang":"en","type":"article","venue":"Journal of Algebra and Its Applications","topic":"Finite Group Theory Research","field":"Mathematics","cited_by":3,"is_retracted":false,"has_abstract":true,"ca_institutions":"Mount Allison University","funders":"","keywords":"Mathematics; Dihedral group; Group (periodic table); Combinatorics; Hyperboloid; Pythagorean triple; Abelian group; Non-abelian group; Cyclic group; Group ring; Order (exchange); Automorphism; Integer (computer science); Pure mathematics; Pythagorean theorem; Geometry","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":[0.0008026417,0.0002924764,0.0005178988,0.001320753,0.002221247,0.002281079,0.0005128163,0.0004676906,0.004836076],"category_scores_gemma":[0.001487272,0.0003137603,0.0004939667,0.001168758,0.004386146,0.005273384,0.002493823,0.001236981,0.0006961553],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0008264844,"about_ca_system_score_gemma":0.0003603562,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0005869575,"about_ca_topic_score_gemma":0.0004788192,"domain_scores_codex":[0.999345,0.0001843142,0.0000520142,0.0001242913,0.0001828202,0.0001113962],"domain_scores_gemma":[0.9995857,0.0001149224,0.0000845574,0.00009579242,0.00004349077,0.00007547455],"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":[0.00002706308,0.000006605003,0.0001572043,0.00001891278,0.000003013836,0.0000696287,0.0003826178,0.0003906005,0.0005437962,0.9931976,0.0002469609,0.004956073],"study_design_scores_gemma":[0.00001204839,0.0000228422,0.0001969946,0.00001197297,0.000006804178,0.0001076656,0.0002677354,0.001294981,0.001241995,0.9916563,0.005170768,0.000009935261],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.7164268,0.002324531,0.1828718,0.001106974,0.0003116656,0.00009500956,0.0002265329,0.0003593545,0.09627733],"genre_scores_gemma":[0.9699897,0.0004830933,0.02232037,0.00008751383,0.0001205515,0.00004815848,0.00008438433,0.00004842236,0.006817746],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.004836076,"threshold_uncertainty_score":0.01617831,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.05646152749253354,"score_gpt":0.3285567428728224,"score_spread":0.2720952153802889,"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."}}