For $r\ge 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$). Find the value of$$\underset{r}{\lim }\frac{h(r)}{r^2}.$$

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

The exact value of [ \lim_{r\to\infty}\frac{h(r)}{r^2} ] is **not known** (this is an Erdős–Graham open problem). ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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