On prime vs. prime power pairs


In this paper, we consider pairs of a prime and a prime power with a fixed difference. We prove an average result on the distribution of such pairs. This is a partial improvement of the result of Bauer (Acta Arith. 85:99–118, 1998).

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  1. 1.

    See the estimate (20) in Sect. 5.

  2. 2.

    Recall that we assume that H is a positive integer.

  3. 3.

    Notice that \(HM^{\frac{1}{k}-1}\gg 1\) by the assumption \(M^{1-\frac{1}{k}}\le H\).

  4. 4.

    Recall that we assume that U is a positive integer.


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The author would like to thank Kohji Matsumoto, Hiroshi Mikawa, Koichi Kawada and Alberto Perelli for their helpful comments and suggestions. This work was supported by Grant-in-Aid for JSPS Research Fellow (Grant Number: JP16J00906).

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Correspondence to Yuta Suzuki.

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Suzuki, Y. On prime vs. prime power pairs. Math. Z. 295, 681–710 (2020). https://doi.org/10.1007/s00209-019-02384-9

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  • Waring–Goldbach problem
  • Circle method

Mathematics Subject Classification

  • Primary 11P32
  • Secondary 11P55