Consider the equation $x^2-a\cdot y^2-b=0$ with $a$ positive and $b$ positive.
It seems that if the square root of a+b is an integer, there are automatically integer solutions for the equation.
But what about if the square root of a+b isn't an integer? It depends: for example, $x^2-2\cdot y^2-4=0$ has a solution, but $x^2-2\cdot y^2-3=0$ has no solutions.
Did I get it right and could someone provide a more precise explanation please?
The set of solutions of $x^2 - 2 y^2 = k$ is invariant under the linear mapping $$ M: \pmatrix{x\cr y\cr} \to \pmatrix{3 x + 4 y\cr 2 x + 3 y\cr}$$ and so is the set of integer solutions.
If $k > 0$, the hyperbola $x^2 - 2 y^2 = k$ has two branches, one in the right half plane and in the left, and the images under positive and negative integer powers of $M$ of the piece of the curve for $x > 0, \, -\sqrt{k/2} \le y \le \sqrt{k/2}$ cover the right branch. Thus there is an integer solution if and only if there is an integer solution with $ |y| \le \sqrt{k/2}$ (by symmetry we can consider $0 \le y \le \sqrt{k/2}$).
If $k < 0$, the branches are in the upper and lower half planes, and we similarly consider the images of the piece of the curve for $y > 0,\, -\sqrt{-k} \le x \le \sqrt{-k}$.