Sum of squares


In the past few posts, I have talked about representing integers as a sum of squares:

In this post, I would like to state Lagrange’s four-square theorem following section 6.4 of Niven-Zuckerman-Montgomery’s An introduction to the theory of number.

Firstly, by applying Hensel’s lemma to the result from the earlier post we get (Theorem 5.14):

Proposition: Let a,b,c be arbitrary integers. Then the congruence ax^2+by^2+cz^2\equiv 0\pmod{p} has a non-trivial solution modulo any prime p.

The theorem stated in the earlier post establishes that there is no need for any condition modulo primes p not dividing abc. The above proposition, application of Hensel’s lemma, just demonstrates it more explicitly by telling that the equation is solvable everywhere locally (i.e. modulo every prime).

Secondly, we need following result from Geometry of numbers (Theorem 6.21):

Minkowski’s Convex Body Theorem for general lattices: Let A be a non-singular n\times n matrix with real elements, and let \Lambda = A\mathbb{Z}^n=\{A\mathbf{s}\in \mathbb{R}^n: \mathbf{s}\in \mathbb{Z}^n\} be a lattice. If \mathcal{C} is a set in \mathbb{R}^n that is convex, symmetric about origin \mathbf{0}, and if \text{vol}(\mathcal{C})> 2^n |\det(A)|, then there exists a lattice point \mathbf{x}\in\Lambda such  that \mathbf{x}\neq 0 and \mathbf{x}\in \mathcal{C}.

Now we are ready to state the theorem (for the proof see Theorem 6.26):

Lagrange’s four-square theorem: Every positive integer n can be expressed as the sum of four squares, n=x_1^2+x_2^2+x_3^2+x_4^2, where x_i are non-negative integers.

2 responses

    • I had planned to discuss the Lagrange’s theorem in detail. But due to lack of time, decided to publish the unfinished version with reference for the proof.

      I enjoy learning about things related to Diophantine equations 🙂