Sur les plus grands facteurs premiers d'entiers consécutifs

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 $p⩽y$. In this paper, firstly we show that the triple consecutive integers with the two patterns $P^{+}(n-1)>P^{+}(n)<P^{+}(n+1)$ and $P^{+}(n-1)<P^{+}(n)>P^{+}(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any $J\in \mathbb{Z},J⩾3$, the $J-$tuple consecutive integers with the two patterns $P^{+}(n+j_0)=\underset{0⩽j⩽J-1}{\min }{P}^{+}(n+j)$ and $P^{+}(n+j_0)=\underset{0⩽j⩽J-1}{\max }{P}^{+}(n+j)$ also have a positive proportion respectively. Secondly for $y=x^{\theta }$ with $0<\theta ⩽1$ 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.

in French

References in corpus (2)

• Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers
• On binary correlations of multiplicative functions

Cited by in corpus (2)

• On binary correlations of multiplicative functions
• Value patterns of multiplicative functions and related sequences