On identities concerning integer parts

TL;DR

This paper generalizes a family of identities related to the integer parts of the golden ratio, originally discovered by Zhuravlev, and explores their properties and characterizations.

Contribution

It extends the simplest identity involving the golden ratio's integer parts, providing new insights into their structure and significance.

Findings

Generalized the simplest identity involving the golden ratio

Identified characterization properties of the golden ratio

Enhanced understanding of integer part identities related to algebraic numbers

Abstract

In 2007 V. Zhuravlev discovered a family of identities concerning integer parts which are satisfied by the number $\frac{\sqrt{5}+1}{2}$. Some of these identities turned out to be characterization properties of the number $\frac{\sqrt{5}+1}{2}$. In this paper we generalize the simplest of these identities.