Autour des plus grands facteurs premiers d'entiers consécutifs voisins d'un entier criblé

Denote by $P^+(n)$ (resp. $P^−(n)$) the largest (resp. the smallest) prime factor of the integer $n$. In this paper, we prove that there exists a positive proportion of integers $n$ having no small prime factor such that $P^+(n) x^{1/3−δ}, 0 < δ \leq 1/12$, where $P_3$ denote the integer having at most three prime factors taken with multiplicity. We also prove that the pattern $P^+(p − 1) < P^+(p + 1)$ holds for a positive proportion of primes under the Elliott-Halberstam conjecture.