### Abstract

Consider the polynomial f(x, y) = xy^{k}+ C for k≥ 2 and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of f(x, y) when x, y≤ H. We also prove a similar result for the k-free values of f(p, q) when p, q≤ 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.

Original language | English |
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Pages (from-to) | 190-207 |

Number of pages | 18 |

Journal | Acta Mathematica Hungarica |

Volume | 149 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 |

Externally published | Yes |

### Fingerprint

### Keywords

- primary 11N32
- secondary 11N37

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On the k-free values of the polynomial xy ^{k}+ C.** / Lapkova, K.

Research output: Contribution to journal › Article › Research › peer-review

^{k}+ C'

*Acta Mathematica Hungarica*, vol. 149, no. 1, pp. 190-207. https://doi.org/10.1007/s10474-016-0594-1

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TY - JOUR

T1 - On the k-free values of the polynomial xyk+ C

AU - Lapkova, K.

PY - 2016

Y1 - 2016

N2 - Consider the polynomial f(x, y) = xyk+ C for k≥ 2 and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of f(x, y) when x, y≤ H. We also prove a similar result for the k-free values of f(p, q) when p, q≤ 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.

AB - Consider the polynomial f(x, y) = xyk+ C for k≥ 2 and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of f(x, y) when x, y≤ H. We also prove a similar result for the k-free values of f(p, q) when p, q≤ 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.

KW - primary 11N32

KW - secondary 11N37

UR - http://www.scopus.com/inward/record.url?scp=84959098275&partnerID=8YFLogxK

U2 - 10.1007/s10474-016-0594-1

DO - 10.1007/s10474-016-0594-1

M3 - Article

VL - 149

SP - 190

EP - 207

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 1

ER -