# A note on Fibonacci number of even index

## TL;DR

This paper establishes a precise criterion involving the golden ratio for identifying Fibonacci numbers with even indices, linking number theory and irrational approximations.

## Contribution

It provides a new characterization of even-indexed Fibonacci numbers using a simple inequality involving the golden ratio and fractional parts.

## Key findings

- The criterion exactly characterizes even-index Fibonacci numbers.
- The result connects Fibonacci numbers with properties of irrational numbers.
- It offers a new perspective on Fibonacci number identification.

## Abstract

We prove that a positive integer $n$ is a Fibonacci number of even index if and only if $\langle n\varphi\rangle+\frac{1}{n}>1$.

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Source: https://tomesphere.com/paper/1705.08305