摘要
As usual, denote by P(n) the largest prime factor of the integer n 1 with the convention P(1) = 1.For 0 < θ < 1, define Tθ(x) := |{p x : P(p-1) ≥ p~θ}|.In this paper, we obtain a new lower bound for Tθ(x) as x →∞, which improves some recent results of Luca et al.(2015) and Chen and Chen(2017). As a corollary, we partially prove a conjecture of Chen and Chen(2017)about the size of Tθ(x).
As usual, denote by P（n） the largest prime factor of the integer n 1 with the convention P（1） = 1.For 0 〈 θ 〈 1, define Tθ（x） ：= ｜{p x ： P（p-1） ≥ p~θ}｜.In this paper, we obtain a new lower bound for Tθ（x） as x →∞, which improves some recent results of Luca et al.（2015） and Chen and Chen（2017）. As a corollary, we partially prove a conjecture of Chen and Chen（2017）about the size of Tθ（x）.
基金
supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission(Grant No.KJ1601213)
