# 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 $$x^2-7y^2=2$$.

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