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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