Abstract
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 Erdős’ conjecture for our specific polynomial. The strongest tool we use is a recent generalization of the determinant method due to Reuss.
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The author is partially supported by OTKA grant no. K104183.
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Lapkova, K. On the k-free values of the polynomial \({xy^k + C}\) . Acta Math. Hungar. 149, 190–207 (2016). https://doi.org/10.1007/s10474-016-0594-1
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DOI: https://doi.org/10.1007/s10474-016-0594-1