{"paper":{"title":"On the largest prime factors less than $y$ of consecutive shifted primes","arxiv_id":"2609.15166","primary_cat":"math.NT","submitted_at":"2026-09-14T07:50:22+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 an integer $n > 1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$.","hedge":{"p":0.75,"label":"asserted"},"own_p":0.05,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.31,"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":"One of Erd\\H{o}s and Tur\\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$.","hedge":{"p":0.78,"label":"speculative"},"own_p":0.02,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.97,"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":0.99,"id":"s2","own":false,"role":"background","text":"Furthermore, Wang conjectures that for $x\\rightarrow\\infty$, one has $\\#\\{p\\leq x:P^+(p-1)<P^+(p+1)\\}\\sim \\pi(x)/2$.","hedge":{"p":0.88,"label":"speculative"},"own_p":0.02,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.9,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.0,"background":0.99,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.99,"id":"s3","own":true,"role":"contribution","text":"In this paper, we show the following results.","hedge":{"p":0.67,"label":"asserted"},"own_p":0.51,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":4,"p":0.99,"id":"s4","own":true,"role":"result","text":"For any $3\\leq y\\ll x^{o(1)}$, for $x\\rightarrow\\infty$, we have \\begin{align*} &\\#\\{n\\leq x:P^+_y(n)<P^+_y(n+1)\\}\\sim \\frac{1}{2}x, \\\\&\\#\\{p\\leq x:P^+_y(p-1)<P^+_y(p+1)\\}\\sim \\frac{1}{2}\\pi(x).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.85,"quantitative":true,"quantitative_p":0.98,"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":5,"p":0.61,"id":"s5","own":false,"role":"contribution","text":"\\end{align*} For any $0<\\alpha<17/32$, let $y=x^\\alpha$.","hedge":{"p":0.51,"label":"asserted"},"own_p":0.36,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.08,"result":0.24,"background":0.03,"limitation":0.04,"future_work":0.0,"contribution":0.61,"interpretation":0.0}},{"i":6,"p":1.0,"id":"s6","own":true,"role":"result","text":"Then there exists $h(\\alpha)>0$ such that \\begin{align*} \\#\\{p\\leq x:P_y^+(p-1)<P_y^+(p+1)\\}\\geq(h(\\alpha)+o(1))\\pi(x).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.21,"quantitative":true,"quantitative_p":0.96,"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":7,"p":0.87,"id":"s7","own":true,"role":"result","text":"\\end{align*} In particular, the function $h$ satisfies $\\lim_{\\alpha\\rightarrow 0^+}h(\\alpha)=1/2$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.12,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.87,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.13}},{"i":8,"p":0.97,"id":"s8","own":true,"role":"result","text":"Similar result also holds for $\\#\\{n\\leq x:P_y^+(n)<P_y^+(n+1)\\}$.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.86,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.97,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.01}},{"i":9,"p":0.72,"id":"s9","own":true,"role":"interpretation","text":"These improve Rivat's result (2001) and Wang's result (2019).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.5,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.48,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.02,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.25,"interpretation":0.72}}]},"compiled":{"main_claim":{"p":1.0,"text":"For any $3\\leq y\\ll x^{o(1)}$, for $x\\rightarrow\\infty$, we have \\begin{align*} &\\#\\{n\\leq x:P^+_y(n)<P^+_y(n+1)\\}\\sim \\frac{1}{2}x, \\\\&\\#\\{p\\leq x:P^+_y(p-1)<P^+_y(p+1)\\}\\sim \\frac{1}{2}\\pi(x).","location":"abstract s4","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 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have \\begin{align*} &\\#\\{n\\leq x:P^+_y(n)<P^+_y(n+1)\\}\\sim \\frac{1}{2}x, \\\\&\\#\\{p\\leq x:P^+_y(p-1)<P^+_y(p+1)\\}\\sim \\frac{1}{2}\\pi(x).","location":"abstract s4"}},"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":10,"sentences_in_abstract":10},"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 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