Primes in solutions to Pell-type equations What is known about primes in solutions to Pell-type equations?
In particular, consider the negative Pell equation $x^2 - 5 y^2 = -1$.
As far as I've been able to check
(in the first $4000$ solutions) the only positive-integer solution with $y$ prime  is $x=38$, $y=17$,
but I don't see any obvious reason why this should be the case.
 A: Let $\alpha = (2 + \sqrt{5})$ and $\beta = (2 - \sqrt{5})$. Note that $\alpha \beta = -1$. The general solution is given by
$$y_n = \frac{\alpha^n - \beta^n}{\alpha - \beta} = \frac{\alpha^n -\beta^n}{2 \sqrt{5}}$$
for $n$ odd. (This follows from general theory and I could explain it but I suspect that you know --- you can use induction, for example). This is related to the fact that $\alpha$ is a unit in the ring $\mathbf{Z}[\sqrt{5}]$. However, what is secretly going on is that there is the larger ring $\mathbf{Z}[\phi]$ where
$$\phi = \frac{1 + \sqrt{5}}{2},$$
and in fact $\phi$ is the fundamental unit, and we have $\alpha = \phi^3$ and $\beta = -\phi^{-3}$. So
$$2 y_n =  \frac{\phi^{3n} - (-\phi)^{-3n}}{\sqrt{5}}$$
is actually divisible by
$$ \frac{\phi^{n} - (-\phi)^{-n}}{\sqrt{5}} = F_n,$$
where $F_n$ is the $n$th Fibonacci number. (This divisibility takes place in $\mathbf{Z}[\phi]$, but the ratio $2y_n/F_n$ is a rational number which is an algebraic integer and thus an actual integer.) Hence $y_n$ is divisible by $F_n$ if $F_n$ is odd and by $F_n/2$ if $F_n$ is even (and the ratio is also $> 1$ for $n > 1$). This shows that $y_n$ is not prime as soon as $F_n >
 2$, so for $n > 3$. Hence $y_3 = 17$ is the only prime value.
For more general Pell-type equations I think one is generally out of luck unless there are forced divisibilities as in this case, and looks similar to the primality or otherwise of the sequence $2^n - 1$.
Added: I guess for those who want a more elementary solution, one can observe (and prove by induction) that the $y$ are given by
$$\frac{1}{2} F_{6n+3} = \frac{F_{2n+1} \cdot (5 F^2_{2n+1}  - 3)}{2}$$
and the RHS is easily seen to be prime only for $n = 1$ whence $F_9/2 = 34/2 = 17$.
