OPEN

Let $p_n$ be the smallest prime $\equiv 1\pmod{n}$ and let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$. Is it true that $p_n>m_n$ for almost all $n$? Does $p_n/m_n\to \infty$ for almost all $n$? Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?

Linnik's theorem implies that $p_n\leq n^{O(1)}$. Erdős [Er79e] writes it is 'easy to show' that for infinitely many $n$ we have $p_n <m_n$.