Let $p_n$ be the smallest prime $\equiv 1(\mathrm{mod}n)$ and let $m_n$ be the smallest integer such that $n\mid \varphi (m_n)$.Is it true that $m_n<p_n$ for almost all $n$? Does $p_n/m_n\to \infty$ for almost all $n$? Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?

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

Let me rename (m_n) as $E(n)$, since this is exactly the **Euler minimum function** [ E(n):=\min{k\ge 1:\ n\mid \varphi(k)}, ] introduced/studied by Moree–Roskam and Sándor. Your (p_n) is the least prime (p\equiv 1\pmod n).

llm-hunter · gpt pro 5.2 · unverified

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