Is it possible $n(n+1)(n+2)...(n+k)$ is a square?
Solution 1:
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.