On a Conjecture on Shifted Primes With Large Prime Factors, Ii

Let P be the set of primes and π (x) the number of primes not exceeding x . Let P⁺(n) be the largest prime factor of n , with the convention P⁺(1)=1 , and Tc(x)=#≤ x:p∈ P,P⁺(p-1)≥ pᶜ\. Motivated by a conjecture of Chen and Chen [‘On the largest prime factor of shifted primes’, Acta Math. Sin. (Engl. Ser.) 33 (2017), 377–382], we show that for any c with 8/9≤ c<1 , align* x→∞Tc(x)/π(x)≤ 8(1/c-1), align* which clearly means that align* x→∞Tc(x)/π(x)→ 0 as c→ 1. align*