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

Kostadinka Lapkova

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

Number Theory 2015-10-21 v2

Authors: Kostadinka Lapkova

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. The strongest tool we use is a recent generalization of the determinant method due to Reuss.

Keywords

k-free values polynomial asymptotic formula determinant method prime numbers number theory integer points

Cite

@article{arxiv.1506.08044,
  title  = {On the $k$-free values of the polynomial $xy^k+C$},
  author = {Kostadinka Lapkova},
  journal= {arXiv preprint arXiv:1506.08044},
  year   = {2015}
}

Comments

Revised version with few inaccuracies corrected. 14 pages

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

Abstract

Keywords

Cite

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