The distribution of powers of primes related to the Frobenius problem

TL;DR

This paper investigates the distribution of prime powers related to the Frobenius problem, providing an asymptotic count of prime powers representable as nonnegative integer combinations of two coprime integers.

Contribution

It extends recent results by deriving an asymptotic formula for the count of prime powers within a Frobenius-related set as one parameter grows large.

Findings

Asymptotic formula for prime power counts as c approaches infinity

Extension of Ding, Zhai, and Zhao's recent results

Quantitative understanding of prime powers in Frobenius problem context

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.