{"paper":{"title":"On the Erd\\H{o}s--Odlyzko--S\\'ark\\\"ozy conjecture for prime-power moduli","arxiv_id":"2610.19597","primary_cat":"math.NT","submitted_at":"2026-10-14T05:12:05+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"Erd\\H{o}s, Odlyzko and S\\'ark\\\"ozy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$. O. Klurman, A. P.","hedge":{"p":0.73,"label":"speculative"},"own_p":0.04,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.67,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"Mangerel and J. Ter\\\"av\\\"ainen (2023) have shown that three primes suffice when $q$ is sufficiently smooth.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.79,"id":"s2","own":true,"role":"contribution","text":"We study the moduli $q=p^\\gamma$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.66,"quantitative":true,"quantitative_p":0.77,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.21,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.79,"interpretation":0.0}},{"i":3,"p":0.95,"id":"s3","own":true,"role":"result","text":"For any $q_\\bullet\\le q$ for which $q_\\bullet\\ge\\exp\\bigl(C(\\log q)^{2/3}(\\log\\log q)^{4/3}\\bigr)$, we give an asymptotic formula for the weighted number of representations of each reduced class as a product of two primes not exceeding $q$ and a third not exceeding~$q_\\bullet$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.39,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.95,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.05,"interpretation":0.0}},{"i":4,"p":1.0,"id":"s4","own":true,"role":"result","text":"We also show that the conjecture holds for all but a proportion $\\exp\\bigl(-c(\\log q)^{1/3}(\\log\\log q)^{-1/3}\\bigr)$ of the reduced classes.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.27,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":0.32,"id":"s5","own":false,"role":"background","text":"Here $C$ and $c$ are positive constants that depend only on $p$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.46,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.06,"result":0.23,"background":0.32,"limitation":0.08,"future_work":0.0,"contribution":0.02,"interpretation":0.29}},{"i":6,"p":0.88,"id":"s6","own":true,"role":"method","text":"Both results are unconditional and they rely on a wide zero-free region with no exceptional zeros for the Dirichlet $L$-functions attached to characters modulo $p^\\gamma$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.89,"main_claim_p":0.14,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.88,"result":0.03,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.09,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.88,"text":"We study the moduli $q=p^\\gamma$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small.","location":"abstract s2","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":5950,"cost_usd_reported":0.0},"compiled_at":"2026-10-15","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.05,"states_limitation":0.79,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.9,"mentions_code_or_data_availability":0.01},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.98,"text":"We study the moduli $q=p^\\gamma$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small.","location":"abstract s2"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":7,"sentences_in_abstract":7},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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