I want to prove that the negative Pell's equation
$$x^2-dy^2=-1$$
is unsolvable in integers if $\sqrt{d} = [a_0, \overline{a_1, \dots, a_n}]$ with $n$ even and minimal (minimal since all $\sqrt{d}$ can be written like this by writing $\sqrt{d} = [a_0, \overline{a_1, \dots, a_n,a_1, \dots, a_n}]$).
This question is more general, asking for no solutions when $n$ is even, and infinitely many when $n$ is odd. I answered the odd case, but I've been struggling to come up with how to do the even case, and am unable to find a source on this.
All I seem to find is the necessary but not sufficient condition that the equation is solvable if $d$ is divisible neither by $4$ nor by a prime of the form $4k+3$ (see here for example) which is not what I'm looking for.
Can someone give me a pointer on how I may show that the equation has no solutions under the condition that the continued fraction expansion has even period?
