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This is a Quantitative Study study.

September 23, 2025Open Access

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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