Proofs that every natural number is a sum of four squares.

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.

Solution 1:

One can easily deduce the four squares theorem from the Gauss-Legendre three squares theorem. The latter is actually less elementary, but there is a very nice proof using Hasse-Minkowski and the Aubry-Davenport-Cassels Lemma.

This is the approach taken in Serre's A Course in Arithmetic, Appendix to Chapter IV. Quadratic forms which satisfy the conclusion of the Aubry-Davenport-Cassels Lemma (ADC forms) have been a topic of recent research of mine, so it is fair to say that this is currently my favorite proof.

Note that the quaternionic proof of Chris Card's answer (which is also a very important one) can be understood and, arguably, clarified in these terms. The sum of four squares form does not itself satisfy the hypotheses of the ADC Lemma. This is related to the fact that it is the norm form of the nonmaximal order $\mathbb{Z}[i,j,k]$ in the Hamilton quaternion algebra $H = \left(\frac{-1,-1}{\mathbb{Q}} \right)$. A maximal order containing it is $\mathbb{Z}[i,j,k,\frac{1+i+j+k}{2}]$. The norm form on this order is the quadratic form

$q = x^2 + y^2 + z^2 + w^2 + xw + yw + zw$.

This form does satisfy the hypothesis of the ADC Lemma (it is Euclidean in my terminology), so it follows immediately from Hasse-Minkowski that it is universal, i.e., represents all positive integers. Moreover the form $q$ is sufficiently closely related to the sum of four squares form that a little elementary fiddling around shows that the sum of four squares form is universal as well. For other examples of this phenomenon, see this recent preprint of R.W. Fitzgerald as well as some papers of J.I. Deutsch that are cited there.

(If you dare, you can take a look at my magnum opus on Euclidean forms and ADC forms. Caveat emptor: more than being unpolished, there are some unsanded sharp corners here.)

Solution 2:

There's a proof in Herstein using quaternions.

[See also Hardy and Wright's An Introduction to the Theory of Numbers. -- PLC.]