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

- Fermat’s two-square theorem
- Legendre’s three-square theorem
- Algorithms for finding the sum of squares representation

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 be arbitrary integers. Then the congruence has a non-trivial solution modulo any prime .

*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 be a non-singular matrix with real elements, and let be a lattice. If is a set in that is convex, symmetric about origin , and if , then there exists a lattice point such that and .

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

Lagrange’s four-square theorem:Every positive integer can be expressed as the sum of four squares, , where are non-negative integers.