{"id":"W2085879054","doi":"10.1007/s00026-012-0139-4","title":"Euler’s Pentagonal Number Theorem and the Rogers-Fine Identity","year":2012,"lang":"en","type":"article","venue":"Annals of Combinatorics","topic":"Analytic Number Theory Research","field":"Mathematics","cited_by":22,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Toronto","funders":"","keywords":"Mathematics; Identity (music); Combinatorics; Euler's formula; Pure mathematics; Discrete mathematics; Mathematical analysis; Physics","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":[0.0008589083,0.0004289915,0.0006612927,0.00172525,0.001450174,0.002402007,0.0005588178,0.0007737395,0.007674847],"category_scores_gemma":[0.002626875,0.0003370246,0.0005754988,0.0008902925,0.003655492,0.005270178,0.002218713,0.002384583,0.001145208],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0008979808,"about_ca_system_score_gemma":0.0004382312,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0007109051,"about_ca_topic_score_gemma":0.0005200097,"domain_scores_codex":[0.9995702,0.0001056162,0.00002606425,0.0001000609,0.000120677,0.00007732281],"domain_scores_gemma":[0.9992255,0.0003746104,0.0000684919,0.0001222861,0.0001067462,0.0001023219],"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":[0.000005585438,0.000002562124,0.00005275687,0.000006156498,0.000001085813,0.00001354068,0.00003530742,0.00008850605,0.00008675976,0.9972247,0.0006480844,0.001835031],"study_design_scores_gemma":[0.000004878833,0.000003344378,0.0001473542,0.000004320145,0.000002222774,0.00006368265,0.00003557867,0.0003085667,0.0001389988,0.9935122,0.005774274,0.000004497049],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"other","genre_gemma":"empirical","genre_scores_codex":[0.1767775,0.003864915,0.1352086,0.009942863,0.001407092,0.00003978609,0.0003124585,0.0002372931,0.6722096],"genre_scores_gemma":[0.9138581,0.001890978,0.02136182,0.001117561,0.0009324773,0.00004416078,0.0001728219,0.00009573633,0.06052615],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.007674847,"threshold_uncertainty_score":0.02567488,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.115915998757068,"score_gpt":0.4068576232975202,"score_spread":0.2909416245404521,"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."}}