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The distribution of powers of primes related to the Frobenius problem

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Abstract

Let 1 < c < d be two relatively prime integers, gc,d = cd − c − d, and let \({\mathbb{P}}\) be the set of primes. For any given integer k ⩾ 1, we prove that

\(\#\left\{{p}^{k}\leqslant{g}_{c,d}:p\in {\mathbb{P}},{p}^{k}=cx+dy,x,y\in {\mathbb{Z}}_{\geqslant0}\right\}\sim \frac{k}{K+1}\frac{{g}^{1/k}}{\mathrm{log}g} \mathrm{as } c\to \infty ,\)

which gives an extension of a recent result of Ding, Zhai, and Zhao.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Enxun Huang & Tengyou Zhu

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  1. Enxun Huang
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  2. Tengyou Zhu
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Correspondence to Tengyou Zhu.

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Huang, E., Zhu, T. The distribution of powers of primes related to the Frobenius problem. Lith Math J (2025). https://doi.org/10.1007/s10986-025-09660-8

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  • DOI: https://doi.org/10.1007/s10986-025-09660-8

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