$p$ is a prime number,$A_{p-1}$ is an array with a length of $p-1$, and the i-th item is $(i!\mod p)\ $($1≤i≤p-1$).$f(p)\ $is the number of different elements present in array $A$.Try to find the upper and lower bounds of $\lim_{p \to \infty}f(p)/p$.
I meet this question in Shuzhimi(a chinese BBS),someone discuss for it.They guess $f(p)/p\sim(1-e^{-1})\ \ \ as\ \ \ {p \to \infty}$.But proving it seems difficult.I calculate the situations when $p≤19$.
$$p=2,f(p)=1$$ $$p=3,f(p)=2$$ $$p=5,f(p)=3$$ $$p=7,f(p)=4$$ $$p=11,f(p)=5$$ $$p=13,f(p)=9$$ $$p=17,f(p)=11$$ $$p=19,f(p)=11$$