Let Unparsed notation: f_\max(n) be the largest $m$ such that $\phi(m) = n$, and Unparsed notation: f_\min(n) be the smallest such $m$, where $\phi$ is Euler's totient function. Investigate Unparsed notation: \max_{n\leq x}\frac{f_\max(n)}{f_\min(n)}.

Let Unparsed notation: f_\max(n) be the largest $m$ such that $\phi(m) = n$, and Unparsed notation: f_\min(n) be the smallest such $m$, where $\phi$ is Euler's totient function. Investigate Unparsed notation: \max_{n\leq x}\frac{f_\max(n)}{f_\min(n)}.

GPT-5.5 Pro (prompted by Price) has proved (see also the comments for a summary) that $$\max_{n\leq x}\frac{f_{\max}(n)}{f_{\min}(n)}=(e^\gamma+o(1))\log\log x.$$

A Lean formalisation of the reduction exists, conditional on Mertens' product theorem and Linnik's theorem; see the formal proof.