On a conjecture on shifted primes with large prime factors

Abstract

Let $$\mathcal {P}$$ be the set of all primes and $$\pi (x)$$ be the number of primes up to x. For any $$n\ge 2$$ , let $$P^+(n)$$ be the largest prime factor of n. For $$0<c<1$$ , let $$\begin{aligned} T_c(x)=\#\left\{ p\le x:p\in \mathcal {P},P^+(p-1)\ge p^c\right\} . \end{aligned}$$ In this note, we prove that there exists some $$c<1$$ such that $$\begin{aligned} \limsup _{x\rightarrow \infty }\frac{T_c(x)}{\pi (x)}<\frac{1}{2}, \end{aligned}$$ which disproves a conjecture of Chen and Chen.