AI 中文总结
研究连续整数最大素因子问题，通过证明满足\(P^+(n)<P^+(n + 1)\)的整数\(n\)的渐近密度大于\(0.280\)进行改进，还得出其他相关结论及\(T_c(x)\)与\(\pi(x)\)关系的上界。
AI 中文摘要
设\(P^+(n)\)表示\(n\)的最大素因子。Erdős和Turán的一个猜想断言，满足\(P^+(n)<P^+(n + 1)\)的整数\(n\)的渐近密度为\(1/2\)。本文证明该密度大于\(0.280\)，改进了吕和王（2025）之前的\(0.2017\)结果。还证明存在正密度的\(n\)使得\(P^+(n)<P^+(n + 1)<x^{41/107 + \varepsilon}\)。对于\(1/2 < c < 1\)，给出了\(T_c(x)\)与\(\pi(x)\)关系的上界。
英文摘要
Let $P^+(n)$ denote the largest prime factor of $n$. One of Erdős and Turán's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is 1/2. In this paper, we prove that this density is larger than 0.280, which improves the previous result 0.2017 by Lü and Wang (2025). We also prove that there exists a positive density of $n$ such that $P^+(n)<P^+(n+1)<x^{41/107+\varepsilon}$. Define $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$. For $1/2<c<1$, we also show that \begin{align*} \mathop{\lim \sup}_{x\rightarrow\infty}\frac{T_c(x)}{π(x)}\leq \min\left(-\frac{7}{2}\log c,\frac{1-δ}{2c}\right), \end{align*} where $δ=δ(c)>0$.
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