{"id":"W2051189409","doi":"10.1515/forum.2008.035","title":"Expander graphs and gaps between primes","year":2008,"lang":"en","type":"article","venue":"Forum Mathematicum","topic":"Graph theory and applications","field":"Mathematics","cited_by":12,"is_retracted":false,"has_abstract":true,"ca_institutions":"Queen's University","funders":"","keywords":"Ramanujan's sum; Mathematics; Expander graph; Combinatorics; Prime power; Discrete mathematics; Chordal graph; Prime (order theory); Graph","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":[],"consensus_categories":[],"category_scores_codex":[0.0002410388,0.000178154,0.0002999888,0.00009591732,0.0003448527,0.0000223501,0.0001674821,0.00008189864,0.0001004671],"category_scores_gemma":[0.0001187555,0.0001462675,0.00009509535,0.0001795187,0.0001764942,0.000120866,0.00007924074,0.0001164224,0.00007276284],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.000007882421,"about_ca_system_score_gemma":0.00001485182,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00000202251,"about_ca_topic_score_gemma":0.000001382381,"domain_scores_codex":[0.9989739,0.00002831928,0.0003169822,0.0002160485,0.0001737751,0.0002909642],"domain_scores_gemma":[0.9988294,0.0004915437,0.0001285941,0.0004089694,0.00003544675,0.0001060788],"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":[0.000002538774,0.00008892668,0.004175611,0.0001215295,0.00004595361,0.000002515779,0.001371922,5.532297e-8,0.0003667682,0.9861428,0.007496979,0.0001844081],"study_design_scores_gemma":[0.0002447606,0.00002793368,0.001338687,0.0000500024,0.00004697539,0.00006535792,0.0004509072,0.00001077989,0.003362864,0.9929355,0.001282927,0.0001832377],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9867264,0.0001107825,0.003160027,0.0004371674,0.00002780358,0.0003778587,0.00001252577,0.000249916,0.008897509],"genre_scores_gemma":[0.982273,0.00002566789,0.01562982,0.0000713076,0.0000269247,0.0000878026,0.000005707556,0.00003436608,0.001845397],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.0124698,"threshold_uncertainty_score":0.5964614,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.05040285597966428,"score_gpt":0.3005628081756454,"score_spread":0.2501599521959811,"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."}}