On a conjecture on shifted primes with large prime factors

TL;DR

This paper proves that for some constant less than one, the proportion of primes p up to x with a large prime factor in p-1 is less than half infinitely often, disproving a previous conjecture.

Contribution

It establishes the existence of a constant c<1 for which the density of primes with large prime factors in p-1 is less than 50%, countering prior conjectures.

Findings

Existence of c<1 with limsup of T_c(x)/π(x) less than 1/2

Disproof of Chen and Chen's conjecture

Quantitative bound on primes with large prime factors in p-1

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 $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.