The Farey Sequence, Stern Brocot Tree and Euclids Theorem

TL;DR

This paper explores properties of Farey's sequence to provide a new proof of Euclid's GCD theorem, connecting rational approximations and number theory concepts.

Contribution

It offers a novel proof of Euclid's GCD theorem leveraging Farey's sequence properties, linking rational approximations with classical number theory.

Findings

Farey's sequence properties can be used to prove Euclid's GCD theorem

The proof provides new insights into rational approximations and gcd relationships

Connections between Farey sequences and classical number theory are elucidated

Abstract

Farey's sequence is a well-known procedure used to generate proper fractions from 0 to 1. Farey sequence is commonly used in rational approximations of irrational numbers, ford circles and in Riemann hypothesis. Thus, in this paper, we aim to use properties of the Farey's sequence to prove the popular gcd theorem.