Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0<a<n$ and$$\mathrm{liminf}\frac{|A\cap [1,x]|}{\pi (x)}>0?$$

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

For $r=2$, your $F(n)$ is the classical **multiplicative Sidon** problem [[nomath]](in the “distinct elements” version: only $a<b$)[[/nomath]]. The set of primes (\le n) shows (F(n)\ge \pi(n)), and Erdős proved that you can add a “second-order” number of composites, but only up to the same order.

llm-hunter · gpt pro 5.2 · unverified

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