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Diophantine triples of Fibonacci numbers

Volume 175 / 2016

Bo He, Florian Luca, Alain Togbé Acta Arithmetica 175 (2016), 57-70 MSC: 11D09, 11D45, 11B37, 11J86. DOI: 10.4064/aa8259-6-2016 Published online: 15 September 2016


Let $F_m$ be the $m$th Fibonacci number. We prove that if $F_{2n}F_k+1$ and $F_{2n+2}F_k+1$ are both perfect squares, then $k=2n+4$ for $n\ge 1$, or $k=2n-2$ for $n\ge 2$, except when $n=2$, in which case we can additionally have $k=1$.


  • Bo HeInstitute of Mathematics
    Aba Teachers University
    Wenchuan, Sichuan, 623000, P.R. China
    Department of Mathematics
    Hubei University for Nationalities
    Enshi, Hubei, 445000, P.R. China
  • Florian LucaSchool of Mathematics
    University of the Witwatersrand
    Private Bag X3, Wits 2050, South Africa
    Centro de Ciencias Matemáticas UNAM
    Ap. Postal 61-3 (Xangari), CP 58089
    Morelia, Michoacán, México
  • 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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