MathOverflow

I believe that interactions between physics and mathematics are widely appreciated today. However, my impression is that many mathematicians still find it difficult to understand the physicists’ style of thinking, particularly the relative lack of rigor...

I've come up with a problem as follow:
Given an integer $k > 1$, a queue $Q$ as a permutation of integers $1$ to $N$. You can apply an operation to the queue as follows:
split the queue into no more than $k$ subqueues by repeatedly moving the front element of the main queue to the back of one of the $k$ subqueues until the main queue is empty....

In analogy with this recent question, I would like to ask about the variant $ \gcd(n^2-1,n!-1)$ since I have just verified that $\gcd(n^2-1,n!-1)=1$ for every integer $2\leq n \leq 20000$.
Question. Does there exist any integer $n>1$ such that $\gcd(n^2-1,n!-1)>1$? If not, can be proved that $n^2-1$ and $n!-1$ are coprime for every integer $n\geq 2$?...