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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?

Thanks.

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    Surprise is relative. Perhaps you might like to ask a more precise question, like "are there any common heuristics which suggest that this is the obvious answer" or "are there elementary techniques which quickly yield this result"?2011-08-27
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    @ Qiaochu Yuan: I seem to recall a general result that when the sum of degrees of a Diophantine equation (here 2/n) is less than 1 you should expect a finite number of solutions, but a quick search doesn't turn it up. Maybe this would answer the question.2011-08-27
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    @Ross: this should follow from Scott Carnahan's answer in the linked MO post, but it only gives the result for fixed $n$.2011-08-27

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