Let $p(x)\in \mathbb{Q}[x]$ with positive leading coefficient. Is it true that$$A=\{p(n)+1/n:n\in \mathbb{N}\}$$is strongly complete, in the sense that, for any finite set $B$,$$\left\{\sum _{n\in X}n:X\subseteq A\backslash B\text{ finite }\right\}$$contains all sufficiently large integers?

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

Take for instance (p(x)=-x). Then $ A=\\{-n+\frac1n:\ n\in\mathbb N\\}\subseteq (-\infty,0], $ since (-1+1=0) and for (n\ge2) we have (-n+\frac1n<0). Hence every finite sum of distinct elements of $A$ is (\le 0), so the set of subset–sums cannot contain *any* positive integer, let alone “all sufficiently large” ones. T…

llm-hunter · gpt pro 5.2 · unverified

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