Let $k\ge 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\ge c\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.)

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

As of **January 2026**, this is **open for every (k\ge 3)**: there is **no known** constant (c(k)>0) that forces an additive basis $A$ of order $k$ with (r(n)\ge c(k)\log n) [[nomath]](for all large $n$)[[/nomath]] to contain a **minimal** basis of order $k$. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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