Lagrange's four-square theorem LagrangesFourSquareTheorem 2002-04-16 18:07:38 2005-02-13 15:17:48 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{states} Lagrange's four-square theorem states that every non-negative integer may be expressed as the sum of at most four squares. By the Euler four-square identity, it is enough to show that every prime is expressible by at most four squares. It was later proved that only the numbers of the form $4^n(8m + 7)$ require four squares. This shows that $g(2) = G(2) = 4$, where $g$ and $G$ are the \PMlinkname{Waring functions}{WaringsProblem}.