论文标题
在质数转移的猜想中，主要因素很大
On a conjecture on shifted primes with large prime factors
论文作者
论文摘要
令$ \ Mathcal {p} $为所有素数的集合，$π（x）$是最高$ x $的总数。对于任何$ n \ ge 2 $，令$ p^+（n）$是$ n $的最大素数。对于$ 0 <c <1 $，让$$ t_c（x）= \＃\ {p \ le x：p \ in \ mathcal {p}，p^+（p-1）\ ge p^c \}。 $ \ limsup_ {x \ rightarrow \ infty} \ frac {t_c（x）} {π（x）} <\ frac12，$$，反驳了陈的猜想。
Let $\mathcal{P}$ be the set of all primes and $π(x)$ 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 $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{π(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.
