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On the $X$-coordinates of Pell equations which are Tribonacci numbers

Volume 179 / 2017

Florian Luca, Amanda Montejano, Laszlo Szalay, Alain Togbé Acta Arithmetica 179 (2017), 25-35 MSC: 11B39, 11J86. DOI: 10.4064/aa8553-2-2017 Published online: 26 May 2017


For an integer $d\geq 2$ which is not a square, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=\pm 1$ which is a Tribonacci number, with a few exceptions that we completely characterize.


  • Florian LucaSchool of Mathematics
    University of the Witwatersrand
    Private Bag X3
    Wits 2050, South Africa
    Department of Mathematics
    Faculty of Science
    University of Ostrava
    30. dubna 22
    701 03 Ostrava 1, Czech Republic
  • Amanda MontejanoFacultad de Ciencias
    UNAM Campus Juriquilla
    Juriquilla, Mexico
  • Laszlo SzalayDepartment of Mathematics and Informatics
    J. Selye University
    Hradna ul. 21
    94501 Komarno, Slovakia
  • Alain TogbéDepartment of Mathematics, Statistics and Computer Science
    Purdue University Northwest
    1401 S, U.S. 421
    Westville, IN 46391, U.S.A.

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