{"paper":{"title":"Density and distribution of quotients associated with Goldring's $w$-function","arxiv_id":"2611.02851","primary_cat":"math.NT","submitted_at":"2026-11-02T09:52:26+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":"For primes $p,q,r$ that are not all equal, let $n=pqr$ and define Goldring's $w$-function by $w(n)=P(p+q)P(p+r)P(q+r)$, where $P(a)$ denotes the largest prime factor of $a$.","hedge":{"p":0.92,"label":"asserted"},"own_p":0.11,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.13,"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":0.99,"id":"s1","own":true,"role":"contribution","text":"In this paper, we resolve three problems of Chen, Shi and Wu.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.34,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":true,"role":"result","text":"As $n\\to\\infty$ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and $w(n)/n$ are $[0,1]$ and $[0,\\infty)$, respectively.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.65,"quantitative":true,"quantitative_p":0.88,"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":3,"p":0.99,"id":"s3","own":true,"role":"result","text":"The same conclusions hold when $n$ is restricted either to $p^2q$ with distinct primes $p,q$ or to squarefree products of three primes.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.16,"quantitative":false,"quantitative_p":0.2,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":4,"p":0.98,"id":"s4","own":true,"role":"result","text":"Nevertheless, on each of these three sets, the proportion of integers $n\\le X$ satisfying $P(w(n))/P(n)>\\alpha$ tends to zero as $X\\to\\infty$, for every fixed $\\alpha>0$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.2,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.02,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":0.87,"id":"s5","own":true,"role":"result","text":"We also determine the complete set of finite joint limit points of the two quotients for $n=p^2q$ with $p\\ne q$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.13,"quantitative":false,"quantitative_p":0.09,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.87,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.13,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.96,"text":"As $n\\to\\infty$ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and $w(n)/n$ are $[0,1]$ and $[0,\\infty)$, respectively.","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":5122,"cost_usd_reported":0.0},"compiled_at":"2026-11-04","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.04,"states_limitation":0.7,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.87,"compares_with_prior_results":0.36,"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":0.87,"text":"In this paper, we resolve three problems of Chen, Shi and Wu.","location":"abstract s1"}},"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":6,"sentences_in_abstract":6},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}