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arXiv · 2208.11316

On a conjecture on shifted primes with large prime factors

Yuchen Ding
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Abstract

Let $\mathcal{P}$ be the set of all primes and $\pi(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\}.$$ In this note, we proved that there exists some $c<1$ such that $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.

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Yuchen Ding. 2022-08-24. On a conjecture on shifted primes with large prime factors. https://arxiv.org/abs/2208.11316

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