Learning

Sums of Squares: From Quadratic Residues to the Two- and Four-Square Theorems

Modular Arithmetic and the Ring Z/nZ → Counting Representations of Four Squares

A rigorous university-level path through the theory of representing integers as sums of squares. Starting from quadratic residues and modular arithmetic, it develops the two-squares theorem via Gaussian integers and the four-squares theorem via quaternions, culminating in explicit representation formulas.