<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] Here is another take.
$x^2-y^2=1$ implies $x^2=y^2+1$.
$x^2=y^2+1 > y^2$ implies $x>y$, that is, $x\ge y+1$.
But then $x^2 \ge (y+1)^2 = y^2+2y+1 \ge y^2+2+1 > y^2+1$, since $y \ge 1$.
Thus, $x^2 > y^2+1$, contradiction.

Another, more succinct way to express this is:
$$
y^2 < y^2+1 < (y+1)^2
$$
Thus, $y^2+1$ is strictly between two consecutive squares and so cannot be a square.
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