Problem 456

Erdős Problem #456: declared status 'open'. Formalized: yes. 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 $m_n<p_n$ 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 $m_n=p$? Current best: Linnik's theorem implies that $p_n\leq n^{O(1)}$. It is trivial that $m_n\leq p_n$ always. van Doorn in the comments has noted that if $n=2^{2k+1}$ then $m_n\leq 2n$ and $p_n\geq 2n+1$. Prize: no. Tags: number theory.