On the largest prime factors of consecutive integers with one integer free of small prime factors

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

Zhiwei Wang (1)

1 IECL - Institut Élie Cartan de Lorraine

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 $P^+(n) < P^+(n + 2)$. Especially, we prove that the pattern $P^+(P_3) < P^+ (P_3 + 2)$ is realized by a positive proportion of $P_3$ with $P^−(P_3) > x^{1/3−δ}, 0 < δ \leq 1/12$, 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.

Domaines: Théorie des nombres [math.NT]

Zhiwei Wang. Autour des plus grands facteurs premiers d'entiers consécutifs voisins d'un entier criblé. 2017. ⟨hal-01647898v2⟩