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Article Dans Une Revue Journal de Théorie des Nombres de Bordeaux Année : 2019

A note on the normal largest gap between prime factors

Gerald Tenenbaum

Résumé

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer~$n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with $n$, we have $$f(n):=\max_{1\leqslant j<\omega(n)}\log \Big({\log p_{j+1}(n)\over \log p_j(n)}\Big)=\log_3n+O(\xi(n))$$ for almost all integers $n$.

Dates et versions

hal-02890673 , version 1 (06-07-2020)

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Citer

Gerald Tenenbaum. A note on the normal largest gap between prime factors. Journal de Théorie des Nombres de Bordeaux, 2019, 31 (3), pp.747-749. ⟨10.5802/jtnb.1107⟩. ⟨hal-02890673⟩
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