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Claim. The answer to Problem 1051 is yes, as the case $d=2$ of Theorem 2(1) of K. Barreto, J. Kang, S.-h. Kim, V. Kovač and S. Zhang, Irrationality of rapidly converging series: a problem of Erdős and Graham, arXiv:2601.21442 (first version 2026-01-29, third version 2026-07-08), recorded on its source card. For $d=2$ the theorem reads: if $a_1\le a_2\le\cdots$ are positive integers with

$$\lim_{n\to\infty}a_n^{1/\phi^n}=\infty,\qquad\phi=\frac{1+\sqrt5}{2},$$

then $\sum_{n=1}^\infty 1/(a_na_{n+1})$ is irrational. The abstract states the form for strictly increasing sequences with $\limsup$ in place of $\lim$; the paper derives it from its Theorem 3, since a strictly increasing sequence has $a_na_{n+1}\ge n^2$ and so meets that theorem's lower bound. The question's hypothesis implies the theorem's: $\liminf a_n^{1/2^n}>1$ gives $a_n\ge c^{2^n}$ for some $c>1$ and all large $n$, so $a_n^{1/\phi^n}\ge c^{(2/\phi)^n}$ tends to infinity because $2>\phi$. Theorem 2(2) shows the rate is sharp: for every $C>1$ there is a strictly increasing integer sequence with $\lim a_n^{1/\phi^n}=C$ whose sum is rational, so Erdős and Graham's request for the strongest theorem of this type is answered as well. The paper has no separate proof of Theorem 2: it proves Theorem 2(1) for $d\ge2$ as the special case with all weights equal to one of its Theorem 3, on weighted products of $d$ consecutive terms, and Theorem 2(2) as an instance of its Theorem 5, so the claim rests on the proofs of Theorems 3 and 5 in Sections 3 and 4. The authors state that the original question was solved autonomously by the agent Aletheia, built on Gemini Deep Think; that they then relaxed the growth condition and generalized the series, as Theorem 2(1); that Gemini Deep Think next found a further generalization and proved it jointly with Aletheia, namely Theorem 3 with every numerator $b_n$ equal to $1$ and with a limit in place of the upper limit in its growth hypothesis, which the authors then generalized by weakening its hypotheses; that the counterexamples and the writing are theirs; and that the paper's theorems and proofs are largely products of human-AI interaction. The agent's own argument for the question is written up separately at Feng and coauthors 2026.

Acceptance. Refereed: the paper is published as Bull. London Math. Soc. 58 (2026), no. 8, article e70457 (the paper link, online 2026-07-29), with the same title and authors; the theorem labels used on this page follow the third arXiv version, whose comment of 2026-07-08 announces the journal acceptance. Reviewed: Thomas Bloom, the site's curator, labels the problem PROVED (LEAN) and credits this paper in the remarks with extending the solution and settling the growth threshold up to the limit rate, restating the golden-ratio theorem and its converse (page last edited 2026-02-01); Bloom is an author of neither paper. Barreto posted the preprint in the problem's forum thread on 2026-01-30; the paper credits the heuristic for the golden-ratio threshold to Terence Tao, citing the site's problem page, and Tao's sketch is Tao's reply of 2026-01-30 in that thread. The Lean formalization posted in the same thread covers the question's original hypothesis, not this theorem, and is recorded on the companion claim page. No independent check of the proof by this project is recorded.

Depends on. No page of this wiki.