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Finiteness results for Diophantine triples with repdigit values

Volume 172 / 2016

Attila Bérczes, Florian Luca, István Pink, Volker Ziegler Acta Arithmetica 172 (2016), 133-148 MSC: Primary 11D61. DOI: 10.4064/aa8089-12-2015 Published online: 23 December 2015

Abstract

Let $g\ge 2$ be an integer and $\mathcal R_g\subset \mathbb{N}$ be the set of repdigits in base $g$. Let $\mathcal D_g$ be the set of Diophantine triples with values in $\mathcal R_g$; that is, $\mathcal D_g$ is the set of all triples $(a,b,c)\in \mathbb{N}^3$ with $c \lt b \lt a$ such that $ab+1$, $ac+1$ and $bc+1$ lie in the set $\mathcal R_g$. We prove effective finiteness results for the set $\mathcal D_g$.

Authors

  • Attila BérczesInstitute of Mathematics
    University of Debrecen
    P.O. Box 12
    H-4010 Debrecen, Hungary
    e-mail
  • Florian LucaSchool of Mathematics
    University of the Witwatersrand
    Private Bag X3, Wits 2050
    Johannesburg, South Africa
    e-mail
  • István PinkInstitute of Mathematics
    University of Debrecen
    P.O. Box 12
    H-4010 Debrecen, Hungary
    and
    University of Salzburg
    Hellbrunnerstrasse 34/I
    A-5020 Salzburg, Austria
    e-mail
  • Volker ZieglerUniversity of Salzburg
    Hellbrunnerstrasse 34/I
    A-5020 Salzburg, Austria
    e-mail

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