Problem Statement
Let $f_{\max}(n)$ be the largest $m$ such that $\phi(m)=n$, and $f_{\min}(n)$ be the smallest such $m$, where $\phi$ is Euler's totient function. Investigate\[\max_{n\leq x}\frac{f_{\max}(n)}{f_{\min}(n)}.\]
Categories:
Number Theory
Progress
Carmichael has asked whether there is an integer $n$ for which $\phi(m)=n$ has exactly one solution, that is, $\frac{f_{\max}(n)}{f_{\min}(n)}=1$. Erdős has proved that if such an $n$ exists then there must be infinitely many such $n$.See also [51].
Source: erdosproblems.com/694 | Last verified: January 16, 2026