# Sur les plus grands facteurs premiers dentiers consécutifs

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1 Analyse IECL - Institut Élie Cartan de Lorraine

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 ∈ Z, J\geq3§, the J−tuple consecutive integers with the two patterns §P^+ n + j 0 = \min {0 \leq j \leq J−1} P^+ n + j and §P^+ n + j 0 = max {0\leq j \leqJ−1} also have a positive proportion respectively. Secondly for §y = x^θ§ with §0 < θ ≤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.

Author: ** Zhiwei Wang - **

Source: https://hal.archives-ouvertes.fr/