New demonstrations about the resolution of numbers into squares

TL;DR

This paper presents new historical demonstrations of the theorem that every positive integer can be expressed as a sum of four squares, including a clear proof based on Euler's theorem, with detailed translation and verification.

Contribution

It provides a new translation and analysis of 18th-century demonstrations of the four squares theorem, highlighting their methods and historical significance.

Findings

New translation of 1774 demonstration

Clear proof based on Euler's theorem

Historical insights into number theory

Abstract

Translated from the Latin original "Novae demonstrationes circa resolutionem numerorum in quadrata" (1774). E445 in the Enestrom index. See Chapter III, section XI of Weil's "Number theory: an approach through history". Also, a very clear proof of the four squares theorem based on Euler's is Theorem 370 in Hardy and Wright, "An introduction to the theory of numbers", fifth ed. It uses Theorem 87 in Hardy and Wright, but otherwise does not assume anything else from their book. I translated most of the paper and checked those details a few months ago, but only finished last few parts now. If anything isn't clear please email me.