Is it possible $n(n+1)(n+2)…(n+k)$ is a square?

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 ?

• Isn't this reducible via completing the square to Pell's equation in the case where $k=1$? – Michael Hardy Sep 21 '11 at 18:11
• @Michael $n(n+1)$ is never a square for $n > 0$. Because $n$ and $n+1$ are relatively prime, the only way for $n(n+1)$ is a square is to that each of $n$ and $n+1$ is individually a perfect square. It is easy to see that this is impossible unless $n=0$. – Srivatsan Sep 21 '11 at 18:14
• (I'll record another argument for $k=1$. Hopefully someone can see a proof for general $k$. :)) The number $n(n+1)$ is strictly between $n^2$ and $(n+1)^2$, and hence is not a perfect square. – Srivatsan Sep 21 '11 at 18:20
• I see. According to Wikipedia: <b>Pell's equation<b> is any Diophantine equation of the form $$x^2-ny^2=1$$ where $n$ is a nonsquare integer. I missed the "nonsquare" part. – Michael Hardy Sep 21 '11 at 19:45
• cant you just use bertrand postulate an the fact that the largest prime less than n+k divides the product? – Robert William Hanks Sep 22 '11 at 11:45

• @JasonDeVito: Correction!! I was completely wrong, you had a very good point. The $l=2$ case is much, much easier and was solved 36 years early in 1939 by Erdos! – Eric Naslund Sep 21 '11 at 18:50