An improvement on the largest prime factors of consecutive integers

Abstract

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 $P^{+}(n)<P^{+}(n+1)<x^{41/107+\epsilon }$. Define $T_c(x):=\#\{p\le x:P^{+}(p-1)\ge p^c\}$. 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 $\delta =\delta (c)>0$.

31 pages