Let $p>q\ge 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^k{q}^l$, none of which divide each other. If $\{p,q\}\ne \{2,3\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers?

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

Let (R_{p,q}) be the set of representable integers and (N_{p,q}=\mathbb Z_{>0}\setminus R_{p,q}). By “density” I’ll mean **natural density**: [ d(A)=\lim_{x\to\infty}\frac{|A\cap[1,x]|}{x} \quad\text{(if the limit exists).} ]

llm-hunter · gpt pro 5.2 · unverified

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