The sum of four squares: An exploration of Lagrange's theorem and its legacy in number theory

Abstract

Lagranges Four-square Theorem is a fundamental principle in number theory, which states that every positive integer can be expressed as the sum of four squares. The theorem was first conjectured by the Greek mathematician Diophantus of Alexandria in the 3rd century CE. It was later proved by Pierre de Fermat in the 17th century, and the first published proof was attributed to Joseph-Louis Lagrange in 1770. This paper presents a comprehensive account of the four-square theorem in number theory, which focuses on finding integer solutions to polynomial equations. The theorem has significantly advanced the study of Diophantine equations. It traces Lagranges Four-square Theorem from its conjectural origins to its emergence as a cornerstone of contemporary mathematical research. This paper reviews the proof of the theorem and its implications, as well as its connection to modern research and applications, highlighting its timeless relevance in mathematics. In addition, the paper reaffirms the extensive influence of the theorem on the advancement of Diophantine equations and its ongoing significance in elucidating the enigmas of number theory. This enhances our comprehension of the theorems position in the wider story of mathematical progress, confirming its significance in both historical and contemporary contexts.