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**Proggy** hmmm, just to make sure I was clear in what I was asking, the original problem would be restated as:

Suppose $\displaystyle n$ is a fixed integer $\displaystyle \neq 0$ and d is a positive integer not a perfect square, and suppose

$\displaystyle x_0^2 - dy_0^2 = n$

with $\displaystyle x_0 \geq 0,\ y_0 \geq 0$. Let $\displaystyle x_1 = ax_0 + bdy_0,\ y_1 = bx_0 + ay_0$, where a>0, b>0 provides a solution to the Fermat-Pell equation, $\displaystyle a^2-db^2=1$. Show that

$\displaystyle x_1^2-dy_1^2=n,\ \ \ \ \ x_1>x_0\geq 0,\ \ \ y_1>y_0\geq 0$.

Conclude that the quation $\displaystyle x^2-dy^2=n$ either has no solutions or infinitely many solutions.