Journal of Prime Research in Mathematics
Vol. 19 (2022), Issue 1, pp. 27 – 33
ISSN: 1817-3462E (Online) 1818-5495 (Print)
ISSN: 1817-3462E (Online) 1818-5495 (Print)
On the solutions of 2x + 2y = z2 in the Fibonacci and Lucas numbers
Hayder R. Hashim
\(^a\)Faculty of Computer Science and Mathematics, University of Kufa, P.O. Box 21, 54001 Al Najaf, Iraq.
Correspondence should be addressed to: : hayderr.almuswi@uokufa.edu.iq
Copyright © 2023 Hayder R. Hashim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Published: Received: 30 September 2022; Accepted: 10 March 2023; Published Online: : 08 June 2023.
Abstract
Consider the Diophantine equation 2x + 2y = z2, 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) ∈ {(Fi, Fj , Fk),(Fi, Fj ,Lk),(Li,Lj ,Lk),(Li,Lj , Fk),(Fi,Lj ,Lk),(Fi,Lj , Fk)} with i, j, k ≥ 1 and Fn and Ln denote the general terms of Fibonacci and Lucas numbers, respectively.
Keywords:
Diophantine equations, exponential Diophantine equations, Fibonacci sequence, Lucas sequence.