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.

Solution 1:

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