A number of years ago, I proved the following result:
For any integer $k$, the number of positive integral solutions to $x(x+1)...(x+n-1) = y^n+k$ with $n \ge 3$ is finite (i.e., there are only a finite number of $(x, y, n)$ satisfying this equation for any $k$).
It is pretty clear that for any fixed $k$ and $n$ there are only a finite number of $x$ and $y$ (you can prove that $y \le |k|$), but the fact that there are only a finite number of $n$ came as a surprise to me. I initially proved that $n < e|k|$ and later derived much stricter bounds.
The way I phrased this is "The product of $n$ consecutive integers is almost never close to an $n$-th power."
My question is whether this result is surprising?