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

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

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.

MAHADI DDAMULIRA¹ FLORIAN LUCA^{2 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

Published

23 September 2020

Final version published online

23 September 2020