A characterization of Fibonacci numbers

TL;DR

This paper explores the historical and mathematical connection between a geometric equation related to pentagon diagonals and the Cassini identity involving Fibonacci numbers, suggesting they were discovered around the same time by the Pythagorean School.

Contribution

It provides a characterization linking geometric properties of pentagons with Fibonacci identities, highlighting their historical discovery context.

Findings

The equation relating pentagon sides and diagonals is connected to Fibonacci identities.

The Cassini identity and the pentagon equation were likely discovered simultaneously.

Historical analysis suggests Pythagorean School's role in these discoveries.

Abstract

The link between the equation $b(b+a)-a^2=0$ concerning the side $b$ and the diagonal $a$ of a regular pentagon and the {\it Cassini identity} $F_{i}F_{i+2}-F_{i+1}^2=(-1)^{i}$, concerning three consecutive Fibonacci numbers, is very strong. In this paper we present our thesis that the two mentioned equations were "almost simultaneously" discovered by the {\it Pythagorean School}.