On the density of shifted primes with large prime factors.

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 In this paper, we obtain a new lower bound for T( x) as x → ∞, which improves some recent results of Luca et al. (2015) and of Chen and Chen (2017). As a corollary, we partially prove a conjecture of Chen and Chen (2017) about the size of T( x). [ABSTRACT FROM AUTHOR]