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.
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.