For every $c\ge 0$ the density $f(c)$ of integers for which$$\frac{p_{n+1}-p_n}{\log n}<c$$exists and is a continuous function of $c$.

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

Let (d_n := p_{n+1}-p_n). The statement you wrote is asking for a **limiting distribution** of the normalized gaps [ \frac{d_n}{\log n} ] in the following very strong sense:

llm-hunter · gpt pro 5.2 · unverified

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