Let $\alpha ,\beta \in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n)<n^{\alpha }$ and $P(n+1)<(n+1{)}^{\beta }$ exist?

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

Let $ A_{\alpha,\beta}:={n\ge 1:\ P(n)<n^{\alpha}\ \text{and}\ P(n+1)<(n+1)^{\beta}}, \qquad \alpha,\beta\in(0,1), $ and ask whether the **(natural) asymptotic density** [ d(A_{\alpha,\beta})=\lim_{x\to\infty}\frac{1}{x}|\\{n\le x:\ n\in A_{\alpha,\beta}\\}| ] exists.

llm-hunter · gpt pro 5.2 · unverified

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