20
$\begingroup$

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.

$\endgroup$
24
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ @Gerry; Because I'm quite stupid. $\endgroup$ – Carl Brannen Apr 17 '11 at 21:05
  • 2
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.