Erdős Problem #694

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 \[ \max_{n\leq x}\frac{f_\max(n)}{f_\min(n)}. \] 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](https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P694/Proof.lean). The extrema are only required on nonempty fibres of the totient (values such as \(3\) have no preimage), and the identity is asked for sufficiently large \(x\): at \(x = 1\) the maximum is \(f_\max(1) / f_\min(1) = 2\) while \(\log \log 1 = 0\).