We know that a necessary but not sufficient condition such that,
$$x^2-dy^2 = -1\tag1$$
is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime factors of $d$ may be both of form $4m+1$ yet is still unsolvable, like,
$$x^2-5\cdot41y^2 = -1\tag2$$
It made me wonder what were the relatives of $(2)$.
Question: Let $p< q$ and be primes of form $4m+1$. Is it true that a necessary (but not sufficient) condition such that
$$x^2-pqy^2 = -1\tag3$$
is not solvable is that $q$ has form $q=u^2+pv^2$?
I checked with the Alpertron for all small prime $q=4m+1$ and the list of unsolvable $q$ starts as,
$$p=5;\;q = 5, 41, 61, 101, 109, 149, 181, 241, 269, 281, 389, 401, 409, 421, 449,\dots$$
which are all of form $u^2+5v^2,$ and
$$p=13;\;q = 13, 17, 29, 53, 61, 113, 157, 181, 269, 313, 337, 373, 389,\dots$$
which are all of form $u^2+13v^2.$
However, $(3)$ can be solvable for some $q=u^2+pv^2$, hence the condition is not sufficient to guarantee non-solvability.
So is the answer true or false? If false, can one give a counter-example of a non-solvable prime pair $p,q$ of form $4m+1$ with $q\neq u^2+pv^2$?