Journal of Prime Research in Mathematics

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.


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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.


Diophantine equations, exponential Diophantine equations, Fibonacci sequence, Lucas sequence.