Suppose $n$ and $m$ are relatively prime integers. Define the symbol (sort of like the Jacobi symbol) U$(n,m)=1$ if and only if each prime $p|n$, there is an integer $k$ such that $n^k = p\pmod m$, and otherwise, U$(n,m)=-1$. This implies if $n$ is prime, then U$(n,m)=1$.
For example, U$(25,8)=-1$ because $5|25$ and there is no such $k$ such that $25^k= 5\pmod 8$ and U$(51,8)=1$ because $3|51$ and $17|51$ and there exist integers $k$ such that $51^k = 3\pmod 8$ and $51^k = 17\pmod 8$.
Suppose $n<m^2$, and $m$ has several (distinct) factors. Do composite numbers $n$ with several factors such that U$(n,m)=1$?
One such example I found is when $m=8!=40320$ and $n=40526683=43*547*1723$.
Since $n = 5083 \pmod m$ and $5083^{37}=43\pmod m$, $5083^{67}=547\pmod m$, and $5083^{89}=1723\pmod m$, U$(n,m)=1$. Does there exist an example of another $n$ with four distinct factors such that U$(n,m)=1$ (using the same $m$)?
