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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

up vote 20 down vote accepted

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
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@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
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@Carl, cheer up, we're all quite stupid - that's why we're here. –  Gerry Myerson Apr 18 '11 at 0:21
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