Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite non-empty $S\subset A$ such that$$\sum _{n\in S}\frac{f(n)}{n}=0?$$What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$?

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

Write [ A_+={n\in A:f(n)=1},\qquad A_-={n\in A:f(n)=-1}. ] Then your condition is equivalent to asking whether there must exist finite, nonempty (P\subseteq A_+) and (N\subseteq A_-) with [ \sum_{n\in P}\frac1n=\sum_{n\in N}\frac1n, ] since (\sum_{n\in S} \frac{f(n)}n=0) is exactly “positive-signed reciprocals = negati…

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

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