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Claim. Theorem 1.1 of Will Cook, Reciprocal-Summable Support Irrationality at Every Integer Base, a note dated July 2026 and first committed to Cook's plectis-erdos repository on 2026-09-11 (the preprint link, pinned), states that if $A$ is an infinite set of positive integers with $\sum_{a\in A}1/a<\infty$, then

$$X_A(b)=\sum_{a\in A}\frac{1}{b^a-1}$$

is irrational for every integer $b\ge2$. At $b=2$ every such $A$ is an instance of Problem 257, answered yes. The note presents the theorem as a written proof of Erdős's 1968 remark that the coprimality hypothesis of his theorem for pairwise coprime supports, the accepted partial claim on its own page, can be removed, and claims neither a new statement nor priority. The argument, in the note's §2: for $N\ge1$ the displacement $\Delta_{b,A}(N)=\sum_{a\in A}(b^{N\bmod a}-1)/(b^a-1)$ equals $(b^N-1)X_A(b)-J_N$ with $J_N$ an integer and is positive, so a rational value $p/q$ would force every displacement to be at least $1/q$; averaging $N$ over the multiples of $Q_t=\operatorname{lcm}(1,\ldots,t)$, each summand's mean is at most $1/a$ and vanishes once $a\mid Q_t$, and the convergence of $\sum1/a$ justifies exchanging the sums, so the mean displacement tends to $0$ as $t\to\infty$, a contradiction. The note's later sections state extensions beyond reciprocal summability, a weighted divisibility criterion and mixed supports, as claimed extensions whose supporting material was not in the public repository at the pinned commit and which it does not present as results.

Covers. Every infinite $A$ with $\sum_{a\in A}1/a<\infty$, at the base $2$ of the question and at every integer base $b\ge2$; this contains the pairwise coprime class of Erdős's 1968 theorem. Not covered: supports with divergent reciprocal sum, the gap the problem's question concerns; the note and the post say that arbitrary infinite supports remain open and that the proposed rational values $1/2$ and $1/21$ are not decided.

Standing. Claimed. Cook posted the note to the problem's discussion thread on 2026-09-11 (the discussion link), writing that Cook got Astra to write up the averaging argument, that AI tools contributed substantially to the research, code and drafting under Cook's direction, that Cook is responsible for the claims, and that the notes have had no independent mathematical review. The note's own authorship statement says that AI agents did most of the research and drafting and that Cook did not independently verify every claim. The post says that the averaging argument is ordinary mathematics and not a Lean theorem, the repository's Lean proofs covering the full-support and pairwise coprime cases only. The site labels the problem OPEN, and no refereed or arXiv version is recorded, so the claim lists no evidence.

Depends on. Nothing in this wiki.