Let, $$f(n,p)=(n+1)(n+2) \cdots (n+p-1)$$ Then show that $f(n,p)$ is a not a perfect square for all $n \in \mathbb{N}$ and for all odd primes $p$.
Consider only the cases when $\mathrm{\gcd}(n,p)=1$.
I have not made any progress in the problem except the trivial case when $\gcd(n,p)=p$. Probably some application of Lagrange's Theorem would solve the problem, but I don't get how.
Erdös and Selfridge showed that the product of $k>1$ consecutive integers is never a $l>1$ power.
Though, I think that there are more elementary means for this problem. My suspicion is that $p$ is an odd prime is a red herring. We simply need to show it for all integers $p \geq 3$. As it turns out, Erdos proved that too, and you can read about it here. It is elementary, but not simple.
I'm wondering if the case of $p-1$ has a special solution.