Product of Consecutive Integers is Not a Power

Question

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.

user642796 · Accepted Answer · 2013-03-09 09:48:46Z

26

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

user642796

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answered Apr 16, 2011 at 18:54

Zev Chonoles

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$\begingroup$ Well, that wasn't the short proof I was expecting. $\endgroup$
– Carl Brannen
Commented Apr 16, 2011 at 21:06
4

$\begingroup$ @Carl, whatever made you expect a short proof? $\endgroup$
– Gerry Myerson
Commented Apr 16, 2011 at 23:46
1

$\begingroup$ @Gerry; Because I'm quite stupid. $\endgroup$
– Carl Brannen
Commented Apr 17, 2011 at 21:05
2

$\begingroup$ @Carl, cheer up, we're all quite stupid - that's why we're here. $\endgroup$
– Gerry Myerson
Commented Apr 18, 2011 at 0:21
1

$\begingroup$ @CarlBrannen - "He who learns must suffer. And even in our sleep pain that cannot forget falls drop by drop upon the heart, and in our own despair, against our will, comes wisdom to us by the awful grace of God." -- Aeschylus $\endgroup$
– Ted Hopp
Commented Mar 27, 2022 at 2:51

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