OPEN
Let $g(n)$ count the number of $m$ such that $\phi(m)=n$. Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that
\[g(n) > n^{1-\epsilon}?\]
Pillai proved that $\limsup g(n)=\infty$ and Erdős
[Er35b] proved that there exists some constant $c>0$ such that $g(n) >n^c$ for infinitely many $n$.
This conjecture would follow if we knew that, for every $\epsilon>0$, there are $\gg_\epsilon \frac{x}{\log x}$ many primes $p<x$ such that all prime factors of $p-1$ are $<p^\epsilon$.
See also [416].
