Are there infinitely many regular primes? We know there are infinitely many irregular ones, and that their percentage should be much smaller than the regular ones, still it is unproven that the latter are infinite.
Let me recall that a prime $p$ is irregular if it divides the class number of $\mathbb{Q}(\zeta_p)$, the cyclotomic field.
Similarly, we cannot prove that there are infinitely many real quadratic fields of class number $1$.
