Sur les plus grands facteurs premiers d'entiers cons\'ecutifs

Abstract

Let $P^+(n)$ denote the largest prime factor of the integer $n$ and $P_y^+(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$. In this paper, firstly we show that the triple consecutive integers with the two patterns $P^+(n-1)>P^+(n) P^+(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any $J\in \mathbb{Z}, J\geqslant3$, the $J-$tuple consecutive integers with the two patterns $P^+(n+j_0)= \min\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ and $P^+(n+j_0)= \max\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ also have a positive proportion respectively. Secondly for $y=x^{\theta}$ with $0<\theta\leqslant 1$ we show that there exists a positive proportion of integers $n$ such that $P_y^+(n)<P_y^+(n+1)$. Specially, we can prove that the proportion of integers $n$ such that $P^+(n)<P^+(n+1)$ is larger than 0.1356, which improves the previous result "0.1063" of the author.