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ON A CONJECTURE ON SHIFTED PRIMES WITH LARGE PRIME FACTORS, II
Part of:
Multiplicative number theory
Published online by Cambridge University Press: 13 September 2024
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 
$$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\rightarrow 0 \quad \text{as } c\rightarrow 1. \end{align*} $$
Keywords
MSC classification
Primary:
11N05: Distribution of primes
Secondary:
11N36: Applications of sieve methods
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
The author is supported by National Natural Science Foundation of China (Grant No. 12201544), Natural Science Foundation of Jiangsu Province, China (Grant No. BK20210784) and China Postdoctoral Science Foundation (Grant No. 2022M710121).
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