{"paper":{"title":"Coprime pairs with equal sum of divisors, assuming smooth shifted primes","arxiv_id":"2610.15674","primary_cat":"math.NT","submitted_at":"2026-10-11T14:56:01+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":0.98,"id":"s0","own":false,"role":"background","text":"Let $h(x)$ be the number of pairs $1\\le a<b<x$ with $\\gcd(a,b)=1$ and $\\sigma(a)=\\sigma(b)$.","hedge":{"p":0.55,"label":"asserted"},"own_p":0.15,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.17,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.98,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":false,"role":"background","text":"Erd\\H{o}s Problem #824 asks whether $h(x)>x^{2-o(1)}$.","hedge":{"p":0.37,"label":"asserted"},"own_p":0.04,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.92,"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":2,"p":0.75,"id":"s2","own":false,"role":"background","text":"By an argument of Pollack and Pomerance, this follows if for every fixed $\\delta>0$ there are $y^{1-o(1)}$ primes $p\\le5y$ with $p+1$ free of prime factors exceeding $y^\\delta$.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.08,"main_claim_p":0.11,"quantitative":true,"quantitative_p":0.85,"role_probabilities":{"gap":0.01,"method":0.12,"result":0.06,"background":0.75,"limitation":0.0,"future_work":0.0,"contribution":0.03,"interpretation":0.03}},{"i":3,"p":0.58,"id":"s3","own":true,"role":"method","text":"We write this implication out with explicit parameters and formalize it in Lean 4.","hedge":{"p":0.92,"label":"asserted"},"own_p":0.93,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.58,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.42,"interpretation":0.0}},{"i":4,"p":0.72,"id":"s4","own":true,"role":"method","text":"We then check that the argument of the recent OpenAI preprint \"Weighted dilation graphs, smooth shifted primes and totient fibers\", which claims the corresponding statement for $p-1$, applies to $p+1$ after a change of sign in two of its sections.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.82,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.2,"role_probabilities":{"gap":0.0,"method":0.72,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.28,"interpretation":0.0}},{"i":5,"p":0.68,"id":"s5","own":true,"role":"limitation","text":"Consequently $h(x)=x^{2-o(1)}$, conditionally on the correctness of that preprint, which has not been refereed and whose main new estimates we have not independently verified.","hedge":{"p":0.51,"label":"hedged_possibility"},"own_p":0.52,"main_claim_p":0.9,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.3,"background":0.0,"limitation":0.68,"future_work":0.0,"contribution":0.0,"interpretation":0.02}}]},"compiled":{"main_claim":{"p":1.0,"text":"Consequently $h(x)=x^{2-o(1)}$, conditionally on the correctness of that preprint, which has not been refereed and whose main new estimates we have not independently verified.","location":"abstract s5","hedge_strength":{"p":0.51,"label":"hedged possibility"}},"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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