ABSTRACT
In 1889, Hermann Minkowski introduced a new method into number theory and started a subject called the geometry of numbers. Minkowski's theorem gives a simple geometric criterion for showing the existence of integers satisfying various conditions and is very useful in proving that certain Diophantine equations have solutions. The French mathematician Albert Girard observed that every prime congruent to 1 mod 4 is a sum of two squares. Fermat, in a letter to Mersenne in 1640, said that he had a proof, and it is generally believed that this is the case. The first published proof is due to Euler in 1747. The proof given is a simple consequence of Minkowski's theorem. Although some mathematicians' devised ways to find solutions, Lagrange was the first to publish a proof that solutions always exist, use Minkowski's theorem to prove solutions exist.
