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There are no Diophantine quadruples of Fibonacci numbers

Volume 185 / 2018

Yasutsugu Fujita, Florian Luca Acta Arithmetica 185 (2018), 19-38 MSC: Primary 11D09; Secondary 11B39. DOI: 10.4064/aa170613-8-12 Published online: 18 June 2018

Abstract

We show that there is no Diophantine quadruple, that is, a set $\{a_1,a_2,a_3,a_4\}$ of four positive integers such that $a_ia_j+1$ is a square for all $1\le i \lt j\le 4$, consisting of Fibonacci numbers.

Authors

  • Yasutsugu FujitaDepartment of Mathematics
    College of Industrial Technology
    Nihon University
    2-11-1 Shin-ei, Narashino, Chiba, Japan
    e-mail
  • Florian LucaSchool of Mathematics
    Wits University
    Private Bag X3
    Wits 2050, South Africa
    and
    Max Planck Institute for Mathematics
    Vivatsgasse 7
    53111 Bonn, Germany
    and
    Department of Mathematics
    Faculty of Sciences
    University of Ostrava
    30. dubna 22
    701 03 Ostrava 1, Czech Republic
    e-mail

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