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 $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^{+}(p-1)\ge p^c\}.$$ In this note, we proved that there exists some $c<1$ such that $$\underset{x\to \infty }{\mathrm{limsup}}\frac{T_c(x)}{\pi (x)}<\frac{1}{2},$$ which disproves a conjecture of Chen and Chen.