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

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Abstract

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.

Original languageEnglish
Pages (from-to)190-207
Number of pages18
JournalActa Mathematica Hungarica
Volume149
Issue number1
DOIs
Publication statusPublished - 2016
Externally publishedYes

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Polynomial
Asymptotic Formula
Determinant
Integer
Generalization

Keywords

  • primary 11N32
  • secondary 11N37

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the k-free values of the polynomial xyk+ C. / Lapkova, K.

In: Acta Mathematica Hungarica, Vol. 149, No. 1, 2016, p. 190-207.

Research output: Contribution to journalArticleResearchpeer-review

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