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arXiv · 2412.18898

The distribution of powers of primes related to the Frobenius problem

Enxun Huang·Tengyou Zhu
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Abstract

Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0} \right\}\sim \frac{k}{k+1}\frac{g^{1/k}}{\log g} \quad (\text{as}~c\rightarrow\infty),$$ which gives an extension of a recent result of Ding, Zhai and Zhao.

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Enxun Huang, Tengyou Zhu. 2024-12-25. The distribution of powers of primes related to the Frobenius problem. https://arxiv.org/abs/2412.18898

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