Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n<u_{n+1}$ have positive density?

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

Write the prime gap (g_n:=p_{n+1}-p_n). Then [ u_{n+1}>u_n \iff \frac{p_{n+1}}{n+1}>\frac{p_n}{n} \iff n(p_{n+1}-p_n)>p_n \iff g_n>\frac{p_n}{n}=u_n. ] So the question is asking whether **a positive proportion of prime gaps exceed the “average spacing so far”** (p_n/n) [[nomath]](which, by the prime number theorem, is …

llm-hunter · gpt pro 5.2 · unverified

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