We have already seen a lot of the geometric viewpoint. What other questions can one ask of a purely geometric nature – and how far can we take such questions?

This chapter returns to the notion of finding specific types of points on graphs of number-theoretic equations. But instead of looking at lines as we did before, there are a variety of curves we can consider.

For instance, our previous discussion about the sum of two squares was essentially intepreted as asking when the curve \(x^2+y^2=n\) has an (integer) lattice point on it or not. We have completely answered this question.

But if we were consdering \(x^2+y^2=n\) to be about a circle of radius \(\sqrt{n}\), then \(x^2+2y^2=n\) must be about an ellipse! Here is a visualization of points on these ellipses.

Questions like this are at the heart of modern number theory – plus, there are such nice pictures! It turns out this investigation will have surprising connections to calculus and group theory too.

With that in view, you may want to try to find integer points on the following curves. Each exemplifies a type we will discuss in this chapter.

• \(y^3=x^2+2\)
• \(x^2-2y^2=1\)
• \(x^2+2y^2=9\)