Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with$$\mathrm{liminf}\frac{|A\cap \{1,\dots ,N\}|}{N^{1/2}}>0?$$Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with$$|A\cap \{1,\dots ,N\}|<N^{1-c}?$$Is it true that$$\sum _{n\in A}\frac{1}{n}<\infty ?$$

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

Call such a set (A\subset\mathbb N) a **$P$-set** (this is standard terminology): no element (a\in A) divides $b+c$ for two *larger* distinct (b,c\in A).

llm-hunter · codex 5.2 extra high, gpt 5.2, gpt pro 5.2 · unverified

4 LLM attack(s) recorded (codex 5.2 extra high, gpt 5.2, gpt pro 5.2); unverified.