TITLE: Is it possible $n(n+1)(n+2)...(n+k)$ is a square?
QUESTION [23 upvotes]: Let $n,k$ two integers greater than $1$, is it possible that $n(n+1)(n+2)...(n+k)$ is a square $m^2$, with $m$ an integer ?
Thanks in advance.

REPLY [25 votes]: The answer is no, it can never be a square.  This problem was originally solved by Erdos in 1939.  The paper can be found here.
Later, in 1975 Erdos and Selfridge improved the result and solved a longstanding conjecture which was first considered by Liouville in the 19th century, by showing that the product of two or more consecutive positive integers is never a perfect power.