solving the equation $X^2-nY^2=d$

Question

we know that equations of the form $X^2-dY^2=Z^2$ have parameter solutions in $\mathbb{Z}$

see here

but what about equations of the form $X^2-nY^2=d$ where $x,y,z,d \in \mathbb{Z}$ ($X$ and $Y$ are variables)

do they have solutions in general? what about special case below:

$X^2-17Y^2=14$

Answer 1

Your particular equation has no integral solutions.

First of all, if $7\mid X$ then $7\mid (X^2 - 14) = 17Y^2$ which implies that $7\mid Y$, but then $7^2\mid (X^2 - 17Y^2) = 14$ but that is false. A similar deduction can be made if $7\mid Y$.

Then $X^2$ is congruent to $1$, $2$ or $4$ $\text{(mod) } 7$ and $-17Y^2$ is congruent to $4Y^2$, so also to $1$, $2$ or $4$ $\text{(mod) } 7$.

That means that $X^2 -17Y^2$ can be congruent to one of the following:

$$ \begin{array}{c|ccc} + & 1 & 2 & 4 \\ \hline 1 & 2 & 3 & 5 \\ 2 & 3 & 4 & 6 \\ 4 & 5 & 6 & 1 \\ \end{array} $$

So $X^2 -17Y^2 \not\equiv 0 \pmod 7$ but $14 \equiv 0 \pmod 7$.

Answer 2

Dickson gives the complete integer solution in this paper.