On the $k$-free values of the polynomial $x{y}^k+C$

Abstract

Consider the polynomial $f(x,y)=x{y}^k+C$ for $k\ge 2$ and any nonzero integer constant $C$. We derive an asymptotic formula for the $k$-free values of $f(x,y)$ when $x,y\le H$. We also prove a similar result for the $k$-free values of $f(p,q)$ when $p,q\le H$ are primes. The strongest tool we use is a recent generalization of the determinant method due to Reuss.

Revised version with few inaccuracies corrected. 14 pages

References in corpus (3)

• Pairs of k-free Numbers, consecutive square-full Numbers
• Power-Free Values of Polynomials
• The density of twins of $k$-free numbers