We have now more or less exhausted a lot of what we can do with linear questions, so we return to other considerations. As a warmup for this and ensuing chapters, consider the following question.

Take a positive integer \(n\) and try to write it as \(n=a^2+b^2\) for \(a,b\in\mathbb{Z}\). For which \(n\) is this possible, for which is it not?

It seems that Albert Girard already knew the answer to this question in the first quarter of the 17th century, and Fermat discovered this a couple years later.

• Girard is an interesting figure, less well-known than his contemporaries; he apparently was the first to use our modern notation for trigonometric functions, and spent his adult life in the Netherlands escaping religious persecution as a Protestant in France.
• Fermat also had a religious side; in the Mathematical Intelligencer number 2 from 2012, classicist David Slavitt translates a moving poem about the dying Christ written in honor of one of Fermat's friends.

A full proof of the answer to this question did not come until Euler (no surprise here) about six score years later.

So try it out! Some things to think about while you try this are:

• Are any special types of numbers easier than others?
• Is there any way of generating new such numbers from old ones?
• If some types of numbers are not a sum of squares, how might you prove this?

A separate question to at least keep track of is how many ways you can write a number as a sum of squares, assuming you can indeed write it in this way?

This chapter is completely devoted to continuing to address questions about writing as a sum of two squares. It will lead us a little far afield, of necessity, to ask (and start to answer) questions about congruences again. Much of it will be devoted to a geometric proof that certain numbers are indeed representable as as sum of two squares. This chapter is a perfect illustration of one of the main themes of this text — the unity of mathematics.