Pell-type Equations

A Pell-type equation in general may not have integer solutions (for example, the equation \(x^2-3y^2=2\)). When it does, it is possible to describe the general solution.

Theorem 4 The equation \(x^2-dy^2=-1\) has an integral solution if and only if there exists \(z_1\in\mathbb{Z}[\sqrt d]\) with \(z_1^2=z_0\).

Consider the general equation \(N(z)=a\). Like in Theorem 1, one can show that all its solutions can be obtained from the solutions \(z\) with \(1\leq z\leq z_0\), where \(z_0\) is the smallest non-trivial solution of Pell’s equation \(N(z)=1\). Thus it is always sufficient to check finitely many values of \(x\). Moreover, there is a simple upper bound for those \(x\).

Theorem 5 If \(a\) is an integer such that the equation \(N(z)=x^2- dy^2=a\) has an integer solution, then there is a solution with \(\displaystyle |x|\leq\frac{z_0+1}{2\sqrt{z_0}}\sqrt{|a|}\) and the corresponding upper bound for \(y=\sqrt{\frac{x^2-a}d}\).

Example Find all integer solutions of \(x^2-7y^2=2\).