Note on the largest prime factors of consecutive integers

In this paper, we give an alternative proof of Tao-Teräväinen’s results by showing \begin{align*} &#{n\leq x:P^+(n-1)<P^+(n)<P^+(n+1)>\max_iP^+(n+a_i)}\gg x,\&#{n\leq x:P^+(n-1)>P^+(n)<P^+(n+1)>\max_iP^+(n+a_i)}\gg x, \end{align*} where the $a_i$ are at most finite in number, with $\mathrm{\mid }{a}_i\mathrm{\mid }=O(1)$. We also obtain some comparison results for five numbers under the Elliott-Halberstam conjecture for friable integers. For any $\epsilon >0$, 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).