Question 5
For each prime $p$, somewhere in the multiverse there exists a kingdom consisting of $p$ islands numbered from $1$ to $p$ with a bridge between any pair of them. When Jana visits a kingdom, coronavirus restrictions mean she must obey the following rule: Directly after visiting island $m$, she can only cross over to island $n$ if
$$
p \mid (m^{2}-n+1)(n^{2}-m+1)
$$
Show that there are infinitely many kingdoms such that Jana cannot travel to every island in this manner.
$$
p \mid (m^{2}-n+1)(n^{2}-m+1)
$$
Show that there are infinitely many kingdoms such that Jana cannot travel to every island in this manner.