# Preuve simplifi\'ee du th\'eor\`eme de Serret sur les nombres \'equivalents

## TL;DR

This paper provides a simpler, purely algebraic proof of Serret's theorem, which characterizes when two irrational numbers are related by the action of PGL(2,Z) based on their continued fractions sharing a common quotient.

## Contribution

It offers a simplified and algebraic proof of Serret's theorem, enhancing understanding of the relation between continued fractions and PGL(2,Z) actions.

## Key findings

- Proves the converse of Serret's theorem using algebraic methods
- Shows that sharing a common quotient in continued fractions implies orbit relation
- Simplifies the existing proof of Serret's theorem

## Abstract

If the continued fractions of two irrational numbers have a common complete quotient, then these two numbers are in the same orbit under the action of $\mathrm{PGL}(2,\mathbb{Z})$. The converse is Serret's well-known theorem, but we give a simpler and purely algebraic proof of it.