Applications to Minkowski's Theorem
Theorem 1 Any convex set (or body) in that has central symmetry and volume greater than contains an integer lattice point other than .
If is an invertible matrix, then is a linear map. The lattice is mapped into a system of points in which we call a lattice , where is equal to the determinant of the lattice .
Theorem 2 Let be any lattice of determinant . Then any convex set symmetrical about the origin whose volume is greater than contains a point of other than .
Example 3 Given any real number and integer , there exists integers such that .
Take -set parallelogram bounded by the four lines
This parallelogram has base , altitude , hence . If we take , then . By Minkowski's fundamental theorem, there must be at least one lattice point other than . Thus
Example 4 Lagrange: Every positive integer can be expressed as a sum of four squares,
In view of the algebraic identity
we see that are sums of four squares, thus is the sum of four squares. Therefore, it is enough to prove the theorem for primes. Let
So we have . Now consider the ball
so it has to be !
Theorem 5 There exists infinitely many positive integers that can be written as a sum of four positive perfect squares, but every integer is a sum of five positive perfect squares.
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