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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

YUCHEN DING Open the ORCID record for YUCHEN DING [Opens in a new window]
YUCHEN DING*
Affiliation:
School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, PR China
*
e-mail: [email protected]
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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

shifted primessieve methodBrun–Titchmarsh inequalityprimes in arithmetic progressionsElliott–Halberstam conjecture

MSC classification

Primary: 11N05: Distribution of primes
Secondary: 11N36: Applications of sieve methods
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 1 , February 2025 , pp. 48 - 55
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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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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