Erdős Problem #694

The problem

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

$$\max_{n\leq x}\frac{f_{\max}(n)}{f_{\min}(n)}$$

(where the maximum is restricted to those $n$ of the form $n=\phi(m)$ for some $m$.)

Statement quoted from erdosproblems.com/694, maintained by Thomas Bloom. Original source: [Er79e,p.80]. Current status there: solved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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