A Diophantine equation is a polynomial equation with integer coefficients in which the variables are required to take integer values. In this chapter and the next, we investigate several types of nonlinear Diophantine equations. In the present chapter we examine the Pythagorean formula, and we also consider the question of which integers can be represented as a sum of two squares. This investigation will involve the Gaussian integers, which are complex numbers that behave much like ordinary integers. We conclude with a proof of Lagrange’s theorem that every integer can be written as a sum of four squares.