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Claim. J. Land, A conditional proof of the irrationality of $\sum_{n\ge1}p_n/2^n$ under a uniform Hardy–Littlewood prime-tuples conjecture, research draft dated 5 September 2026, nine pages, in the author's GitHub repository beetree/math_erdos_251 (the preprint link, pinned to the repository's last commit of 2026-09-06). Its Theorem 2 states that Conjecture 1 implies

$$\sum_{n\ge1}\frac{p_n}{2^n}\notin\mathbb Q,$$

where $2=p_1<p_2<\cdots$ are the primes, so the series is the constant of Problem 251 exactly. The argument passes to the prime-gap tails $G_n=\sum_{j\ge0}(p_{n+j+1}-p_{n+j})2^{-j-1}$, which a rational value would force to lie in a fixed lattice, and uses the hypothesis to find indices along which $G_n>6$ and $G_n\to6$, through consecutive primes in the quadratic pattern $t+2j^2$ followed by a long empty interval. The source card land_2026_conditional_proof_irrationality_prime_series holds the digest and its result page Theorem 2 gives the hypothesis, the statement and the claimed proof as the paper presents them; this outline is a reading aid, not proof coverage.

Hypothesis. The claim is conditional on the paper's Conjecture 1, a large-$x$ form of Kuperberg's Conjecture 1.3 (conjecture_1_3): for some absolute $\varepsilon,C>0$ and all large $x$, every admissible set $A\subseteq[0,(\log x)^2]$ of at most $(\log\log x)^3$ shifts has its prime-tuple count up to $x$ within $Cx^{1-\varepsilon}$ of $\mathfrak S(A)\int_2^x(\log y)^{-|A|}\,dy$. The conjecture is unproved; the paper itself says that the lower counts it needs already imply infinitely many twin primes. Acceptance of the claim would establish an implication, not the irrationality, and would leave the problem open.

Postings. The author announced the draft on the problem's discussion thread on 2026-09-06 and, the same day, the Lean development of the repository, whose terminal theorem derives Irrational of the site's series from a Prop encoding the conjecture (the formalization link, at the same pinned commit). No proof claim was registered on the site's proof-claims page. The paper states that it was prepared with the assistance of the AI systems gpt-6-astra, fable 5.1 and gemini-3.8-flash, that the author is responsible for its content, and that no proof-assistant verification is claimed for the paper; the repository's README reports an author-run build and axiom audit of the conditional theorem, which the corpus has not reproduced.

Standing. Claimed. The draft is not refereed and not on arXiv, no outside reader is recorded as having reviewed it, the site's curator has not acknowledged it, and the card digests its statements and outline, not its proof; the author's Lean development is not part of this repository's audited Lean, so it gives no formalized evidence, and no Lean statement of the repository has been compared with the manuscript. Ringer claims the same implication by a different argument from a different specialization of the same conjecture, and states that no implication between the two specialized hypotheses is claimed.

Depends on. Nothing in this wiki.