On a conjecture on shifted primes with large prime factors, II
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910834891948032 |
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| author | Ding, Yuchen |
| author_facet | Ding, Yuchen |
| contents | Let $\mathcal{P}$ be the set of primes and $π(x)$ the number of primes not exceeding $x$. Let also $P^+(n)$ be the largest prime factor of $n$ with convention $P^+(1)=1$ and $$ T_c(x)=\#\left\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\right\}. $$ Motivated by a 2017 conjecture of Chen and Chen, we show that for any $8/9\le c<1$ $$ \limsup_{x\rightarrow\infty}T_c(x)/π(x)\le 8(1/c-1), $$ which clearly means that $$ \limsup_{x\rightarrow\infty}T_c(x)/π(x)\rightarrow 0, \quad \text{as}~c\rightarrow1. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09829 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a conjecture on shifted primes with large prime factors, II Ding, Yuchen Number Theory Let $\mathcal{P}$ be the set of primes and $π(x)$ the number of primes not exceeding $x$. Let also $P^+(n)$ be the largest prime factor of $n$ with convention $P^+(1)=1$ and $$ T_c(x)=\#\left\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\right\}. $$ Motivated by a 2017 conjecture of Chen and Chen, we show that for any $8/9\le c<1$ $$ \limsup_{x\rightarrow\infty}T_c(x)/π(x)\le 8(1/c-1), $$ which clearly means that $$ \limsup_{x\rightarrow\infty}T_c(x)/π(x)\rightarrow 0, \quad \text{as}~c\rightarrow1. $$ |
| title | On a conjecture on shifted primes with large prime factors, II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.09829 |