Original scientific paper
https://doi.org/10.64785/mc.30.2.2
On the Diophantine equation \(F_{n}^{2}+F_{m}^{2}=2^{\alpha}\)
Ga-Hyun Jeong
; Department of Mathematics Education, Kyungnam University, Republic of Korea
Poo-Sung Park
; Department of Mathematics Education, Kyungnam University, Republic of Korea
Abstract
Let \(F_{n}\) be the 𝑛-th Fibonacci number. In this paper, we show that the only nonnegative solutions (𝑛, 𝑚, 𝑎)
of the Diophantine equation \(F_{n}^{2}+F_{m}^{2}=2^{\alpha}\) with 𝑛 ≥ 𝑚 ≥ 0 are (1, 0, 0), (2, 0, 0), (3, 0, 2), (6, 0, 6), (1, 1, 1), (2, 1, 1), (2, 2, 1), (3, 3, 3), (6, 6, 7).
Keywords
Fibonacci numbers; Diophantine equation; linear forms in logarithms; reduction method
Hrčak ID:
335660
URI
Publication date:
22.9.2025.
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