Note on shifted primes with large prime factors
Yuchen Ding, Zhiwei Wang
TL;DR
This paper investigates primes $p\le x$ with the property that the largest prime factor of $p-1$ is at least $p^c$, quantified by $T_c(x)$. It advances the unconditional bound on the proportion of such primes by combining a Rosser–Iwaniec linear sieve with a Bombieri–Friedlander–Iwaniec type distribution result, yielding $\limsup_{x\to\infty} T_c(x)/\pi(x)\le -\frac{7}{2}\log c$ for $e^{-{2/7}}<c<1$; this improves prior bounds that required $c$ to be closer to $1$ and broadens the applicable range of $c$. The approach also yields corollaries such as $\limsup_{x\to\infty} T_c(x)/\pi(x)<1/2$ for $c>e^{-1/7}$, representing a substantial sharpening of previous estimates. The results highlight the effectiveness of refined sieve methods and level of distribution estimates in understanding shifted primes with large prime factors and have implications for related problems in prime gaps and arithmetic progressions. All key constants and bounds are explicit up to $o(1)$ terms, and the methods bridge classical sieve techniques with modern distribution results.
Abstract
For any $0<c<1$ let $$ T_c(x)=|\big\{p\le x: p\in \mathbb{P}, P^+(p-1)\ge p^c\big\}|, $$ where $\mathbb{P}$ is the set of primes and $P^+(n)$ denotes the largest prime factor of $n$. Erd\H os proved in 1935 that $$ \limsup_{x\rightarrow \infty}T_c(x)/π(x)\rightarrow 0, \quad \text{as~}c\rightarrow 1, $$ where $π(x)$ denotes the number of primes not exceeding $x$. Recently, Ding gave a quantitative form of Erd\H os' result and showed that for $8/9< c<1$ we have $$ \limsup_{x\rightarrow \infty}T_c(x)/π(x)\le 8\big(c^{-1}-1\big). $$ In this article, Ding's bound is improved to $$ \limsup_{x\rightarrow \infty}T_c(x)/π(x)\le -\frac{7}{2}\log c $$ for $e^{-\frac{2}{7}}< c<1$.