OPEN
This is open, and cannot be resolved with a finite computation.
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\[\max_{n\leq x}\frac{f_{\max}(n)}{f_{\min}(n)}.\]
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].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #694, https://www.erdosproblems.com/694, accessed 2026-01-16
