Question 41
For each prime $p$, the kingdom of $p$-Landia consists of $p$ islands numbered $1,2,\ldots,p$. Two distinct islands numbered $n$ and $m$ are connected by a bridge if and only if
$$
p\mid(n^2-m+1)(m^2-n+1).
$$
Prove that for infinitely many primes $p$, there are two islands in $p$-Landia that cannot be connected by a chain of bridges.
$$
p\mid(n^2-m+1)(m^2-n+1).
$$
Prove that for infinitely many primes $p$, there are two islands in $p$-Landia that cannot be connected by a chain of bridges.