A famous theorem of Lagrange says that every positive integer is a sum of 4 perfect squares (some may be 0). More subtle is sums of 3 squares, because not every positive integer has that form. For example, 7, 15, and more generally numbers of the form 8k+7 are not a sum of 3 squares (of integers).

Often in number theory, the problem of studying integer solutions to an equation lies much deeper than the study of rational solutions (because rational solutions are more susceptible to geometric methods). Amazingly, the 3 square problem is the same for rational numbers as it is for integers: if an integer is a sum of 3 rational squares then it is automatically also a sum of 3 integral squares! After discussing a little history, we will show the nifty argument that proves this fact.