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 ∈ 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.
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Zhiwei Wang. Sur les plus grands facteurs premiers d'entiers consécutifs. 2017. ⟨hal-01501694⟩

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