Note on shifted primes with large prime factors
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908815154216960 |
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| author | Ding, Yuchen Wang, Zhiwei |
| author_facet | Ding, Yuchen Wang, Zhiwei |
| contents | We denote by $P^+(n)$ the largest prime factor of the integer $n$. In 1935, Erd\H os studied the quantity $T_c(x)$ defined by
$$
T_c(x)=\big|\big\{p\le x: P^+(p-1)\ge p^c\big\}\big|,
$$
and he proved
$$
\limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\rightarrow 0, \quad \text{as~}c\rightarrow 1.
$$
Recently, Ding gave a quantitative form of Erd\H os' result, showing that
$$
\limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\le 8\big(c^{-1}-1\big).
$$
holds for $8/9< c<1$. In this paper, we improve Ding's upper bound to
$$
\limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\le -\frac{7}{2}\log c
$$
for $e^{-\frac{2}{7}}< c<1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04026 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Note on shifted primes with large prime factors Ding, Yuchen Wang, Zhiwei Number Theory We denote by $P^+(n)$ the largest prime factor of the integer $n$. In 1935, Erd\H os studied the quantity $T_c(x)$ defined by $$ T_c(x)=\big|\big\{p\le x: P^+(p-1)\ge p^c\big\}\big|, $$ and he proved $$ \limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\rightarrow 0, \quad \text{as~}c\rightarrow 1. $$ Recently, Ding gave a quantitative form of Erd\H os' result, showing that $$ \limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\le 8\big(c^{-1}-1\big). $$ holds for $8/9< c<1$. In this paper, we improve Ding's upper bound to $$ \limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\le -\frac{7}{2}\log c $$ for $e^{-\frac{2}{7}}< c<1$. |
| title | Note on shifted primes with large prime factors |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.04026 |