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.
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.