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.


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

  • $\begingroup$ Well, that wasn't the short proof I was expecting. $\endgroup$ – Carl Brannen Apr 16 '11 at 21:06
  • 4
    $\begingroup$ @Carl, whatever made you expect a short proof? $\endgroup$ – Gerry Myerson Apr 16 '11 at 23:46
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    $\begingroup$ @Gerry; Because I'm quite stupid. $\endgroup$ – Carl Brannen Apr 17 '11 at 21:05
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    $\begingroup$ @Carl, cheer up, we're all quite stupid - that's why we're here. $\endgroup$ – Gerry Myerson Apr 18 '11 at 0:21
  • $\begingroup$ @Gerry; Actually, the miserable people I know are all quite smart. $\endgroup$ – Carl Brannen Apr 18 '11 at 0:24

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