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On the Diophantine equation \ (F₍^2+F₌^2=2^\) | Synapse

September 23, 2025Open Access

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On the Diophantine equation \ (F₍^2+F₌^2=2^\)

Key Points

The study identifies specific nonnegative solutions of the diophantine equation involving Fibonacci numbers.
Only a limited set of pairs of Fibonacci indices yields valid integer solutions.
Analysis explores the structural properties of Fibonacci numbers in the context of mathematical equations.
Findings can inspire further research on relationships between Fibonacci numbers and integer solutions for similar equations.

Abstract

Let ???????? be the ????-th Fibonacci number. In this paper, we show that the only nonnegative solutions (????, ????, ????) of the Diophantine equation ???? 2 ???? + ???? 2 ???? = 2 ???? with ???? ≥ ???? ≥ 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)

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Cite This Study

Jeong et al. (Mon,) studied this question.

synapsesocial.com/papers/68d4758931b076d99fa6d0fc https://doi.org/https://doi.org/10.64785/mc.30.2.2

Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

1On a Diophantine Equation with Jacobsthal and Fibonacci Numbers2026
2A classification of Markoff-Fibonacci m-triples2024
3Some Diophantine equations involving arithmetic functions and Bhargava factorials2024
4On Tribonacci numbers written as a product of three Fibonacci numbers2026
5Sufficient conditions for solvability of linear Diophantine equations, and Frobenius numbers2024