Archimede, Theon de Smyrne et $\sqrt3$

TL;DR

This paper explores Theon of Smyrna's algorithm for approximating square roots, its historical significance, and its connections to Archimedes' inequalities, Pell-Fermat equations, and continued fractions.

Contribution

It reveals the potential prior knowledge of Greek mathematicians like Archimedes regarding the algorithm used for approximating square roots and related number theory concepts.

Findings

Theon of Smyrna's algorithm applies to √3 and yields Archimedes inequalities.

The algorithm may have been known to ancient Greeks before Theon.

Connections to Pell-Fermat equations and continued fractions are discussed.

Abstract

We know that the algorithm of Theon of Smyrna (70-135 AD) made it possible to highlight fine frames of $\sqrt2$ by rationals. However, this same algorithm also applies to $\sqrt3$ and makes it possible to find the famous Archimedes inequalities. An interesting question is whether this very simple iterative method exposed by Theon was not known to the Greeks before him, notably by Archimedes or his contemporaries. Knowing that Theon of Smyrna had already compiled previous works. Note that this algorithm had also made it possible to open a breach towards the Pell-Fermat equations and continued fractions.