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reviewer:erdos-db-trust · no key registered on this bundle

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null → 9f75542b

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in state · unreviewed

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1. 2026-05-30proposeproposed · finding.addagent:erdos-spine-ingestvpr_65bc6ffad7c2a755Candidate claim vc_cb7973c232de2315 imported from artifact packet cap_61973ee16b553d57
2. 2026-05-30acceptfinding.assertedreviewer:erdos-db-trustnull→9f75542bvev_bbe9c37600e79fc2Candidate claim vc_cb7973c232de2315 imported from artifact packet cap_61973ee16b553d57

Erdős Problem #198 has status 'disproved (lean)'. Statement: The answer is no; Erdős and Graham report this was proved by Baumgartner, presumably referring to the paper [Ba75], which does not state this exactly, but the following simple construction is implicit in [Ba75]. Let $P_1,P_2,\ldots$ be an enumeration of all countably many infinite arithmetic progressions. We choose $a_1$ to be the minimal element of $P_1\cap \mathbb{N}$, and in general choose $a_n$ to be an element of $P_n\cap \mathbb{N}$ such that $a_n>2a_{n-1}$. By construction $A=\{a_1 < a_2 < \cdots\}$ contains at least one element from every infinite arithmetic progression, and is a lacunary set, so is certainly Sidon. AlphaProof has found the following explicit construction: $A = \{ (n+1)!+n : n\geq 0\}$. This is a Sidon set, and intersects every arithmetic progression, since for any $a,d\in \mathbb{N}$, $(a+d+1)!+(a+d)\in A$, and $d$ divides $(a+d+1)!+d$. This was formalized in Lean by Alexeev using Aristotle. Topics: additive combinatorics, sidon sets, arithmetic progressions. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.