It is also impossible to find any integer $n$ such that $$ n^2 \equiv 2 \pmod 3. $$
EDITTTTT: Note that if you have a positive prime $p \equiv 1 \pmod 4,$ then there is always an integer solution to $$ x^2 - p y^2 = -1.$$ Proof in Mordell's book. For prime coefficients there is thus no ambiguity, as for positive prime $q \equiv 3 \pmod 4,$ there is never an integer solution to $$ x^2 - q y^2 = -1.$$
It starts to get tricky with composite coefficients. There is no integer solution to $x^2 - 34 y^2 = -1.$ There is no integer solution to $x^2 - 205 y^2 = -1.$ Here 205 is the smallest odd number that gives a surprise. And, when I say surprise, note that there is never a solution to $x^2 - k^2 y^2 = -1$ when $k >1.$