Three conjectures on P+(n) and P+(n + 1) hold under the Elliott-Halberstam conjecture for friable integers

Abstract

Denote by P+(n) the largest prime factor of an integer n. In this paper, we show that the Elliott-Halberstam conjecture for friable integers (or smooth integers) implies three conjectures concerning the largest prime factors of consecutive integers, formulated by Erdős-Turán in the 1930s, by Erdős-Pomerance in 1978, and by Erdős in 1979 respectively. More precisely, assuming the Elliott-Halberstam conjecture for friable integers, we deduce that the three sets E1={n⩽x:P+(n)⩽xs,P+(n+1)⩽xt}, E2={n⩽x:P+(n)<P+(n+1)xα}, E3={n⩽x:P+(n)<P+(n+1)} have an asymptotic density ρ(1/s)ρ(1/t), ∫Tαu(y)u(z)dydz, 1/2 respectively for s,t∈(0,1), where ρ(⋅) is the Dickman function, and Tα, u(⋅) are defined in Theorem 2.