$\large \color{#3D99F6}{x}^2-\color{#D61F06}{y}^2=2011$

How many integral solutions $(\color{#3D99F6}{x},\color{#D61F06}{y})$ are there for the equation above?

The answer is 4.

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As $2011$ is prime so $x-y=1$ and $x+y=2011\Rightarrow 2x=2012 \therefore x=1006$ consequently $y=1005$ .

But as we are talking of square numerbs $x=\pm 1006$ and $y=\pm 1005$ . So there are $\boxed{4}$ solutions.

$(1006,1005),(-1006,1005),(1006,-1005),(-1006,-1005)$