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.

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 $\langle n\varphi\rangle+\frac{1}{n}>1$.