On a conjecture on shifted primes with large prime factors, II

Abstract

Let $\mathcal{P}$ be the set of primes and $\pi (x)$ the number of primes not exceeding $x$. Let also $P^{+}(n)$ be the largest prime factor of $n$ with convention $P^{+}(1)=1$ and $$T_c(x)=\#\left\{p\le x:p\in \mathcal{P},P^{+}(p-1)\ge p^c\right\}.$$ Motivated by a 2017 conjecture of Chen and Chen, we show that for any $8/9\le c<1$ $$\underset{x\to \infty }{\mathrm{limsup}}{T}_c(x)/\pi (x)\le 8(1/c-1),$$ which clearly means that $$\underset{x\to \infty }{\mathrm{limsup}}{T}_c(x)/\pi (x)\to 0,\text{as}c\to 1.$$

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