# Number Theory Seminar

Tuesday, October 25, 2022 — 2:30 PM EDT

Gian Cordana Sanjaya, Department of Pure Mathematics, University of Waterloo

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

A classical question in analytic number theory is to determine the density of integers $a_1, \ldots, a_n$ such that $P(a_1, \ldots, a_n)$ is squarefree, where $P$ is a fixed integer polynomial. In this talk, we consider the case $P(a, b) = a^4 + b^3$. When the pairs $(a, b)$ are ordered by $\max\{|a|^{1/3}, |b|^{1/4}\}$, we prove that this density equals the conjectured product of local densities. More generally, we prove the same result for $P(a, b) = \beta a^4 + \alpha b^3$, where $\alpha$ and $\beta$ are fixed nonzero integers such that $\gcd(\alpha, \beta)$ is squarefree. This is joint work with Xiaoheng Wang.

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