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On the density of shifted primes with large prime factors

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Abstract

As usual, denote by P(n) the largest prime factor of the integer n ⩾ 1 with the convention P(1) = 1. For 0 < θ < 1, define

$${T_\theta }\left( x \right): = \left| {\left\{ {p \leqslant x:P\left( {p - 1} \right) \geqslant {p^\theta }} \right\}} \right|.$$

In this paper, we obtain a new lower bound for T θ (x) as x → ∞, which improves some recent results of Luca et al. (2015) and of Chen and Chen (2017). As a corollary, we partially prove a conjecture of Chen and Chen (2017) about the size of T θ (x).

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Acknowledgements

This work was supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1601213). This work was begun during Workshop on Analytic Number Theory at Xi’an Jiaotong University, China, August 18–20, 2016. The authors thank Xi’an Jiaotong University for agreeable hospitality and Yonggao Chen for his excellent talk on the largest prime factors of shifted primes, which initialized this work.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yangtze Normal University, Chongqing, 200240, China

    Bin Feng & Jie Wu

  2. CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, 54506, France

    Jie Wu

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  1. Bin Feng
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  2. Jie Wu
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Correspondence to Jie Wu.

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Feng, B., Wu, J. On the density of shifted primes with large prime factors. Sci. China Math. 61, 83–94 (2018). https://doi.org/10.1007/s11425-016-9065-7

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  • DOI: https://doi.org/10.1007/s11425-016-9065-7

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