Let $f_{\max }(n)$ be the largest $m$ such that $\varphi (m)=n$, and $f_{\min }(n)$ be the smallest such $m$, where $\varphi$ is Euler's totient function. Investigate$$\underset{n\le x}{\max }\frac{f_{\max }(n)}{f_{\min }(n)}$$(where the maximum is restricted to those $n$ of the form $n=\varphi (m)$ for some $m$.)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let [ \Phi^{-1}(n):={m\ge 1:\varphi(m)=n},\qquad f_{\min}(n)=\min\Phi^{-1}(n),\quad f_{\max}(n)=\max\Phi^{-1}(n), ] and define [ R(x):=\max_{\substack{n\le x\ \Phi^{-1}(n)\neq\varnothing}}\frac{f_{\max}(n)}{f_{\min}(n)}. ] [[nomath]](This is exactly Erdős problem #694, and it is listed as open. $[Erdős Problems][1]$)[[…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.