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    • The $x$-coordinates of Pell equations and sums of two Fibonacci numbers II

      MAHADI DDAMULIRA FLORIAN LUCA

      Article Volume 130 Published: 23 September 2020 Article ID 0058

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      https://www.ias.ac.in/article/fulltext/pmsc/130/0058

    • Keywords

       

      Fibonacci number; Pell equation; linear form in logarithm; reduction method.

    • Abstract

       

      Let $\{Fn\}_{n≥0}$ be the sequence of Fibonacci numbers defined by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for all $n\geq 0$. In this paper, for an integer $d \geq 2$ which is square-free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2 − dy^2 = \pm 4$ which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize.

    • Author Affiliations

       

      MAHADI DDAMULIRA1 FLORIAN LUCA2 3 4

      1. Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24/II, 8010 Graz, Austria
      2. School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
      3. Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia
      4. Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico

    • Dates

       
      Published
      23 September 2020
      Final version published online
      23 September 2020

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