{"id":"W2012306397","doi":"10.1515/integ.2011.058","title":"Reducing the Erdős–Moser Equation 1 n + 2 n + ⋯ + kn = (k + 1) n Modulo k and k 2","year":2011,"lang":"en","type":"preprint","venue":"Integers","topic":"Advanced Mathematical Identities","field":"Mathematics","cited_by":4,"is_retracted":false,"has_abstract":true,"ca_institutions":"TD Bank Group","funders":"","keywords":"Mathematics; Divisibility rule; Congruence (geometry); Modulo; Diophantine equation; Congruence relation; Quotient; Fermat's Last Theorem; Conjecture; Combinatorics; Corollary; Pure mathematics; Functional equation; Discrete mathematics; Mathematical analysis; Differential equation","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":[0.0007932281,0.0006280955,0.0007387794,0.0009423183,0.001490651,0.001442114,0.00105927,0.001120666,0.006734369],"category_scores_gemma":[0.002416242,0.0002436314,0.0007553149,0.000626969,0.001981012,0.003755594,0.002525595,0.00257202,0.001106437],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001156758,"about_ca_system_score_gemma":0.0008535758,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001027252,"about_ca_topic_score_gemma":0.001300176,"domain_scores_codex":[0.9993566,0.0001672005,0.00004141666,0.0001268992,0.0001924934,0.0001153955],"domain_scores_gemma":[0.9996455,0.0001167903,0.00007091385,0.00005688441,0.00006041476,0.0000495736],"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":[0.00008672567,0.00003545115,0.0003114423,0.00009233291,0.00001432569,0.0003382976,0.0004551044,0.002275505,0.003226946,0.9755836,0.005014362,0.01256598],"study_design_scores_gemma":[0.00005571235,0.00003423525,0.0002581893,0.00002239181,0.00001300811,0.0002488862,0.0001982071,0.004911696,0.001785632,0.982188,0.01026381,0.00002011696],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.6979816,0.001909551,0.06933466,0.01250459,0.002769881,0.0001096255,0.0002879889,0.0002497916,0.2148524],"genre_scores_gemma":[0.9125496,0.001615094,0.02820902,0.001714932,0.001261589,0.00008152533,0.0002341097,0.0001309193,0.05420325],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.006734369,"threshold_uncertainty_score":0.02252871,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.147077676913408,"score_gpt":0.3448385526522958,"score_spread":0.1977608757388878,"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."}}