Note on the largest prime factors of consecutive integers

Abstract

In this paper, we give an alternative proof of Tao-Teräväinen's results by showing \begin{align*} &\#\{n\leq x:P^+(n-1) \max_iP^+(n+a_i)\}\gg x,\\&\#\{n\leq x:P^+(n-1)>P^+(n) \max_iP^+(n+a_i)\}\gg x, \end{align*} where the $a_i$ are at most finite in number, with $|a_i| = O(1)$. We also obtain some comparison results for five numbers under the Elliott-Halberstam conjecture for friable integers. For any $\varepsilon>0$, we also show \begin{align*} &\#\left\{n\leq x:1\leq \frac{P^+(n+1)}{P^+(n)}<1+\varepsilon\right\}\gg_\varepsilon \frac{x}{\log x}, \end{align*} which improves a previous result $\gg x(\log x)^{-50}$ due to S. Kim (2020).