# Product of Consecutive Integers is Not a Power

Is it true that the product of $n>1$ consecutive integers is never a $k$-th power of another integer for any $k \geq 2$?

I can see this is true in certain cases. For instance if the product ends on a prime, But how would one prove this in general?

Thanks for any help or suggestions.

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## 1 Answer

Yes, this is true. This was proven by Erdős and Selfridge in this paper.

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Thanks for that quick and exact response! – pel Apr 16 '11 at 19:04
Well, that wasn't the short proof I was expecting. – Carl Brannen Apr 16 '11 at 21:06
@Carl, whatever made you expect a short proof? – Gerry Myerson Apr 16 '11 at 23:46
@Gerry; Because I'm quite stupid. – Carl Brannen Apr 17 '11 at 21:05
@Carl, cheer up, we're all quite stupid - that's why we're here. – Gerry Myerson Apr 18 '11 at 0:21