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-Turan 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 E 1 = { n ⩽ x : P + ( n ) ⩽ x s , P + ( n + 1 ) ⩽ x t } , E 2 = { n ⩽ x : P + ( n ) P + ( n + 1 ) x α } , E 3 = { n ⩽ x : P + ( n ) P + ( n + 1 ) } have an asymptotic density ρ ( 1 / s ) ρ ( 1 / t ) , ∫ T α u ( y ) u ( z ) d y d z , 1/2 respectively for s , t ∈ ( 0 , 1 ) , where ρ ( ⋅ ) is the Dickman function, and T α , u ( ⋅ ) are defined in Theorem 2.