On a Conjecture on Shifted Primes With Large Prime Factors, Ii
Year: 2024
Type: article
Abstract: Let $\mathcal {P}$ be the set of primes and $\pi (x)$ the number of primes not exceeding x. Let $P^+(n)$ be the largest prime factor of n, with the convention $P^+(1)=1$ , and $ T_c(x)=\#\{p\le x:p\in \mathcal {P},P^+(p-1)\ge p^c\}. $ Motivated by a conjecture of Chen and Chen [‘On the largest prime factor of shifted primes’, Acta Math. Sin. (Engl. Ser.)33 (2017), 377–382], we show that for any c with $8/9\le c<1$ , $$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\le 8(1/c-1), \end{align*} $$ which clearly means that $$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\rightarrow 0 \quad \text{as } c\rightarrow 1. \end{align*} $$
Author: YUCHEN DING
Institution: Yangzhou University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,450
Awards: G4874119299, G6701958676, G910499618
OpenAlex ID: W4402521891