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On the solutions of 2x + 2y = z2 in the Fibonacci and Lucas numbers

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Title: On the solutions of 2x + 2y = z2 in the Fibonacci and Lucas numbers

Authors: Hayder R. Hashim

Journal: Journal of Prime Research in Mathematics

HEC Recognition History
Category From To
Y 2023-07-01 2024-09-30
Y 2022-07-01 2023-06-30
Y 2021-07-01 2022-06-30
Y 2020-07-01 2021-06-30

Publisher: Abdus Salam School of Mathematical Sciences, GC University

Country: Pakistan

Year: 2023

Volume: 19

Issue: 1

Language: English

Keywords: Diophantine equationFibonacci numbersLucas numbers

Categories
Fibonacci Numbers (10) Lucas Numbers (2) Pell's Equation (1) Number Theory (108) Diophantine Equations (4)

Abstract

Consider the Diophantine equation 2<sup>x</sup> + 2<sup>y</sup> = z<sup>2</sup>, where x, y and z are nonnegative integers. As thisequation has infinitely many solutions, in this paper we study its solutions in case where the unknowns represent Fibonacci and/or Lucas numbers. In other words, we completely resolve the equation in case of (x, y, z) ∈ {(F<sub>i</sub>, F<sub>j</sub> , F<sub>k</sub>),(F<sub>i</sub>, F<sub>j</sub> ,L<sub>k</sub>),(L<sub>i</sub>,L<sub>j</sub> ,L<sub>k</sub>),(L<sub>i</sub>,L<sub>j</sub> , F<sub>k</sub>),(F<sub>i</sub>,L<sub>j</sub> ,L<sub>k</sub>),(F<sub>i</sub>,L<sub>j</sub> , F<sub>k</sub>)} with i, j, k ≥ 1 and F<sub>n</sub> and L<sub>n</sub> denote the general terms of Fibonacci and Lucas numbers, respectively.

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