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Ho-Hon Leung,
On squares of Fibonacci numbers and Lucas numbers of the form (p^s-1)^x + p^y.
Int. J. Math. Comput. Sci., 21, no. 1, (2026), 175-188.

DOI:

https://doi.org/10.69793/ijmcs/01.2026/leung

Keywords and phrases:

Exponential Diophantine equation, Fibonacci number, Lucas number.

Abstract:

Let F_k be the Fibonacci numbers and L_k be the Lucas numbers. In this paper, we study the exponential Diophantine equations F_k^2 = (p^s -1)^x + p^y and L_k^2 = (p^s -1)^x + p^y, where s>0, x,y \geq 0 and p is a prime which is congruent to 3 modulo 4. We also obtain partial results for the first equation when the prime p is congruent to 1 modulo 4.