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On the number of pairs of positive integers $x, y \leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free

Volume 194 / 2020

Stoyan Dimitrov Acta Arithmetica 194 (2020), 281-294 MSC: Primary 11L05, 11N25; Secondary 11N37. DOI: 10.4064/aa190118-25-7 Published online: 6 March 2020

Abstract

We show that there exist infinitely many consecutive square-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also establish an asymptotic formula for the number of pairs of positive integers $x, y \leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free.

Authors

  • Stoyan DimitrovFaculty of Applied Mathematics and Informatics
    Technical University of Sofia
    Blvd. St. Kliment Ohridski 8
    1756 Sofia, Bulgaria
    e-mail

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