Prove that $(n-2)(n-1)n(n+1)(n+2)$ is never a perfect square for $n \ge 3$.
I've the following progress: $\gcd(n^2-1,n^2-4) = \gcd(3,n^2-4)$ which is either $1,3$.
When the $\gcd$ is $1$ it's trivial as $n^2-1$ would have to be a perfect square.
Now when the $\gcd$ is $3$ is where I am stuck...
I tried with cases based on what $\gcd(n^2-4,n)$ is: $1,2$ or $4$.
If it is $1$ the problem is easy, but I haven't been able to prove it for when the $\gcd$ is $2$ or $4$.
I would appreciate any help to prove this.
