Erdos 281 solved with ChatGPT 5.2 Pro

Re:Erdos 281 solved with ChatGPT 5.2 Pro by Davidzheng on Sunday January 18, 2026 @03:05AM (#46665779)

"Since $U_{k+1} \subseteq U_k$, the sets $U_k$ are decreasing and periodic, and their intersection $U = \bigcap_{k \ge 1} U_k$ has density $d = \lim_{k \to \infty} d_k \ge \epsilon$."

Is this enough? Let $U_k$ be the set of integers such that their remainder mod 6^n is greater or equal to 2^n for all 1<n<k. Density of each $U_k$ is more than 1/2 I think but not the intersection (empty) right?

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Re:Erdos 281 solved with ChatGPT 5.2 Pro by Paracompact on Sunday January 18, 2026 @05:15AM (#46666505)

Indeed. Your sets are decreasing periodic of density always greater than the product from k=1 to infinity of (1-(1/3)^k), which is about 0.56, yet their intersection is null.

This would all be a fairly trivial exercise in diagonalization if such a lemma as implied by Deepseek existed.

(Edit: The bounding I suggested may not be precise at each level, but it is asymptotically the limit of the sequence of densities, so up to some epsilon it demonstrates the desired counterexample.)

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