<s>[INST] How to prove that there are no positive integer solutions $(x, y)$ to the equation $x^2 - y^2 = 1$ Prove the following:
Theorem. There are no positive integer solutions $(x, y)$ to the equation
$x^2-y^2=1$.
 [/INST] If $x^2 -y^2=1$, then $(x-y) = \frac{1}{x+y} $, since $x$ and $y$ are positive integers $0<\frac{1}{x+y}<\frac{1}{2}$. But since x and y are integers so does $x-y$. Which gives us a contradiction.
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