Find all positive integer such that $$\phi(n)^2\mid n^2-1$$
My progress so far:
I proved that $n$ has to be a odd square-free integer. So $n=p_{1}p_{2}\ldots p_{k}$. So the question can be restated as following: $$\prod (p_{i}-1)^2\mid (p_{1}p_{2}\ldots p_{k})^2-1$$ From here I farther transformed the question as follows (Not sure if this simplifies the problem but it is definitely beautiful):
If $P(x)$ a polynomial such that all it's roots are distinct odd primes, prove that $$P(1)^2\nmid P(0)^2-1$$
And now I am stuck. Any help is appreciated.