An equation of the form \(x^2 - dy^2 = 1\) where \(d>0\) is a nonsquare (what if \(d\) is a square?) integer and we seek \(x,y\) in the integers is called Pell’s equation. By changing the sign of \(x\) and \(y\), we may assume they are nonnegative. There is always the solution \((x,y)=(1,0)\) which we call trivial.
Here is a useful way to think about the solutions to Pell’s equations. The difference of squares identity prompts us to consider the solution \((x,y)\) to Pell’s equation as the real number \(x+y\sqrt{d}\). The fact that \((x,y)\) is a solution simply means \((x+y\sqrt{d})(x-y\sqrt{d})=1\).
Here is some common notation you might see if you look at books or on the internet: if we have a number \(u=x+y\sqrt{d}\), we can define its conjugate as the number \(\bar u=x-y\sqrt{d}\). Moreover, the set of numbers of the form \(x+y\sqrt{d}\) where \(x,y\) are whole numbers, is often written as \(\mathbb Z[\sqrt{d}]\) (pronounced “integers adjoint \(\sqrt{d}\)"). Therefore, a number \(u\) that belongs to \(\mathbb Z[\sqrt{d}]\) is a solution to Pell’s equation if \(u\bar u=1\).
As with all Diophantine equations, we would like to to know the following about Pell’s equation.
Does Pell’s equation always have nontrivial solutions?
When Pell’s equation does have solutions, is the number of solutions finite?
How can we describe all solutions to Pell’s equation?
In this sheet, we answer all of the questions above and apply these theoretical results to some other problems.