On the k-free values of the polynomial {xy^k + C}

Consider the polynomial \({f(x, y) = xy^k + C}\) for \({k \geq 2}\) and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of \({f(x, y)}\) when \({x, y \leq H}\). We also prove a similar result for the k-free values of \({f(p, q)}\) when \({p, q \leq H}\) are primes, thus extending Erd?s’ conjecture for our specific polynomial. The strongest tool we use is a recent generalization of the determinant method due to Reuss.