On the squarefree values of $a^4+b^3$

Abstract

In this article, we prove that the density of integers $a, b$ such that $a^4+b^3$ is squarefree, when ordered by $\max\{|a|^{1/3},|b|^{1/4}\}$, equals the conjectured product of the local densities. We show that the same is true for polynomials of the form $\beta a^4 + \alpha b^3$ for any fixed integers $\alpha$ and $\beta$. We give an exact count for the number of pairs $(a,b)$ of integers with $\max\{|a|^{1/3},|b|^{1/4}\}<X$ such that $\beta a^4 + \alpha b^3$ is squarefree, with a power-saving error term.