We have been doing a lot of stuff now with squares. It is almost time to see one of the great theorems of numbers, which gives us great insight into the nature of squares in the integer world – and whose easiest proof involves lattice points!

This theorem, in the next chapter, will come from our trying to find the solution to a useful general problem, which I like to think of as the last piece of translating high school algebra to the modular world. That is the task of solving quadratic congruences, the modular equivalent to the well-known quadratic equations.

A quadratic congruence is just something of the form \begin{equation*}ax^2+bx+c\equiv 0\text{ (mod }n)\end{equation*} In algebra, we would use the quadratic formula. This chapter will see how far we can extend this to the modular world.