It is not "B\'ezout's identity"

TL;DR

This paper clarifies Euclid's original work on what is now called Bézout's identity, arguing that the common attribution to Bézout is historically inaccurate and exploring Euclid's true approach.

Contribution

It provides a detailed analysis of Euclid's arguments related to gcd and linear combinations, correcting the misattribution of Bézout's identity and discussing historical misconceptions.

Findings

Euclid's work predates Bézout's and contains the original ideas.

Misattribution has led to confusion in historical and modern contexts.

Euclid's proofs are rooted in geometric interpretations of number theory.

Abstract

Given two non-zero integers $a$ and $b$ there exist integers $m$ and $n$ for which $am-bn =(a,b)$. An increasing number of mathematicians have been calling this `B\'ezout's identity', some encouraged by finding "identit\'e de B\'ezout" in Bourbaki's \emph{'El\'ements de math\'ematique}. Moreover the observation that if $\gcd(a,b)=1$ then this is an `if and only if' condition, is sometimes called the "Bachet-B\'ezout theorem". However this is all in Euclid's work from around 300 B.C., when his writings are interpreted in context. So why does he not get credit? Some authors learned the name "B\'ezout's identity" and have perhaps not consulted Euclid, so copied the misattribution. Others, like some Nicolas Bourbaki collaborators, have perhaps browsed Euclid's results, but in a form written for the modern mathematician, and missed out on what he really did (though certainly others, such as Weil, did not). In this article we will carefully explain what Euclid's arguments are and what his approach was. We will also share Kowalski's guess as to the reasons behind Bourbaki's misnomer. To appreciate Euclid, you need to read his work in context: Lengths are the central object of study to the geometer Euclid, though he brilliantly developed the theory of the numbers that measured those lengths. Today's mathematicians read his number theory results as being about abstract numbers not measurements. However the correct interpretation changes how these results are perceived; Euclid's proofs make clear Euclid's intentions. These misperceptions reflect recent discussions about difficulties faced by indigenous people when learning mathematics. We discuss how some indigenous groups may learn numbers in certain practical contexts, not as abstract entities, and struggle when curricula assume that we all share abstract numbers as a basic, primary fully-absorbed working tool.