"Exceptional" primes greater than 2

Question

In many theorems, the prime $2$ has an exceptional characteristic. But where can other primes be exceptional? As a couple examples:

1) In the Fibonacci sequence $5$ is exceptional because $p=5$ allows $p|F_p$, unlike other primes where instead $p|F_{p+1}$ or $p|F_{p-1}$.

2) Pandiagonal Latin squares have both $2$ and $3$ as exceptional primes because none exist when the order of the square is divisible by $2$ or $3$.

Answer 1

Here is one possible non-trivial and exceptional prime, the existence or non-existence of which is not yet proven.

It is believed, but not yet proven, that there does not exist positive integers $m$ and $n$ for which $\zeta(1+m/n)$ is a positive integer. However when it comes to prime we can prove a slightly better result i.e.

There exists either none or at most only one prime $p$ such that $\zeta(1+p/n)$ is an positive integer for some natural number $n$.

I believe there is no such prime, but if we are wrong then there will be at most one such prime, making this prime truly exceptional.