2005/06/03 by D. A. Goldston, S. W. Graham, Goldston, D. A. +6 · 1 citation
Mathematics · #11N25 (primary) 11N05 #11N36 (secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.NT #msc:11N05 #msc:11N25 #msc:11N36
paper · pdf · doi:10.48550/arxiv.math/0506067
49 pages
arxiv created 2005/06/03 · openalex publication_date 2005/06/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let pn denote the nth prime. Goldston, Pintz, and Yildirim recently proved that \liminfn→ ∞ \frac(pn+1-pn)log pn =0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that \liminfn→ ∞ (qn+1-qn) ≤ 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.