TITLE: Proofs that every natural number is a sum of four squares.
QUESTION [22 upvotes]: I am planning to write a little note detailing several proofs of Lagrange's theorem that every natural number can be written as the sum of four perfect squares. I know of three different proofs so far:

a completely elementary proof by descent. 
a proof via Minkowski's theorem and lattices. 
Jacobi's proof via modular forms. 

Can anybody think of any more nice, relatively elementary proofs of this result? Thanks in advance.

REPLY [10 votes]: Perhaps the most beautiful solution is by way of Aubry's lemma - which employs a geometric variant of the Euclidean algorithm to turn a rational represention into an integral representation. This is the same technique that leads to the reflective generation of primitive Pythagorean triples and the associated ternary tree structure. Aubry's results are, in fact, very special cases of general results of Wall, Vinberg, Scharlau et al. on reflective lattices, i.e. arithmetic groups of isometries generated by reflections in hyperplanes. Generally reflections generate the orthogonal group of Lorentzian quadratic forms in dim < 10. See my MO post here for further remarks and references.
In my opinion, the results in this area are some of the most beautiful results in elementary number theory. Strangely enough, for all this beauty they appear to be little known. For example, over a decade ago when I mentioned to John Conway the connection between Aubry's work and Cassels and Pfister he was not aware of this (R.K. Guy told me that the presentation of the PPT ternary tree in their Book of Numbers (1996) is based on a lecture he heard by an undergraduate, Richard Vogeler, at an MAA section at Brigham Young Univ. on 89-04-07.)  Also Pfister apparently was not aware of Aubry's work when he generalized Cassels result to arbitrary quadratic forms, founding the modern algebraic theory of quadratic forms ("Pfister forms"). Someday I hope to write something on the bizarre history of this beautiful circle of ideas so I would be grateful to hear from anyone who may know further details.