Barreto 2026 irrationality rapidly converging series problem erdos

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theorem_2: For psi the root above one of psi^d = psi^(d-1) + 1, a non-decreasing positive-integer sequence with a_n^(1/psi^n) tending to infinity makes the sum of 1/(a_n a_{n+1} ... a_{n+d-1}) irrational, while for every C > 1 some strictly increasing sequence with a_n^(1/psi^n) tending to C makes it rational; the paper reads the case d = 2 as a positive answer to its Question 1, which is Problem 1051.

theorem_3: For non-negative integer weights w_0, ..., w_{d-1} with w_{d-1} >= 1 and c_w the positive root of (x-1) sum w_j x^j - W x^(d-1), W the largest weight, the sum of b_n over the weighted product of a_n, ..., a_{n+d-1} is irrational when b_n and the products obey polynomial bounds and a_n^(1/c_w^n) has limit superior infinity.

theorem_5: For non-negative integer weights with w_{d-1} >= 1 and c~w the largest positive root of (x-1) sum w_j x^j - x^(d-1), every C > 1 admits a strictly increasing positive-integer sequence with a_n^(1/c~w^n) tending to C and a rational sum of 1/(a_n^{w_0} ... a{n+d-1}^{w{d-1}}); with 0-1 weights this shows Theorem 3 is sharp.

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Kevin Barreto, Jiwon Kang, Sang-hyun Kim, Vjekoslav Kovač, Shengtong Zhang, Irrationality of rapidly converging series: a problem of Erdős and Graham. arXiv:2601.21442 (2026); the arXiv comment on v3 says the paper is to appear in the Bulletin of the London Mathematical Society.

Answering a question of Erdős and Graham, Theorem 2(1) shows that if a monotonically increasing sequence of positive integers satisfies lim a_n^{1/psi^n} = infinity, where psi is the positive root of x^d = x^{d-1} + 1, then the sum of 1/(a_n a_{n+1} ... a_{n+d-1}) is irrational; for d = 2 the exponent psi is the golden ratio, so the Erdős–Graham hypothesis lim inf a_n^{1/2^n} > 1 suffices. Theorem 2(2) shows sharpness: for every C > 1 there is a strictly increasing sequence with lim a_n^{1/psi^n} = C whose corresponding sum is rational, so for d = 2 the condition lim inf a_n^{1/phi^n} > 1 is not enough. Theorem 3 generalizes the positive result to weighted Cantor-type series sum b_n / (a_n^{w_0} ... a_{n+d-1}^{w_{d-1}}) with the critical growth rate given by the root c_w of (x-1) sum w_j x^j - W x^{d-1}, W the largest weight, recovering Erdős's earlier theorem on sum 1/a_n as the case d = 1; Theorem 5 supplies a negative statement for the largest root of (x-1) sum w_j x^j - x^{d-1}, which coincides with c_w, so that Theorem 3 is sharp, when every weight w_j is 0 or 1. The method combines partial-sum denominator versus tail size estimates (the heuristic c^d - 1 <= c^{d-1} that produces the critical exponent, sketched by Tao for the golden ratio) with explicit perturbation constructions for the rational counterexamples. The paper is also notable methodologically: the original question was solved autonomously by the AI agent Aletheia built on Gemini Deep Think, with the generalizations produced by human-AI collaboration. It bears directly on problem 1051, which asks exactly Question 1 of the paper, and answers it affirmatively; the paper offers Theorem 2 as its answer to Erdős and Graham's request for the strongest theorem of this type.

Source: https://arxiv.org/abs/2601.21442. The arXiv record (https://arxiv.org/abs/2601.21442, read 2026-10-02) names the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 license. The copy read for this card is arXiv:2601.21442v3 (8 Jul 2026).

Bears on. #1051 (the paper's Question 1 quotes the problem; by the paper's remark on p. 3, Theorem 2(1) with $d=2$ gives the irrationality of $\sum1/(a_na_{n+1})$ under the problem's hypothesis $\liminf a_n^{1/2^n}>1$, an affirmative answer, and Theorem 2(2) with $d=2$ shows that the hypothesis $\liminf a_n^{1/\phi^n}>1$ does not suffice)

Results.

• Theorem 2 (pp. 2--3): for a non-decreasing sequence of positive integers with $\lim a_n^{1/\psi^n}=\infty$, where $\psi^d=\psi^{d-1}+1$, the sum of $1/(a_na_{n+1}\cdots a_{n+d-1})$ is irrational; for every $C>1$ some strictly increasing sequence with $\lim a_n^{1/\psi^n}=C$ makes it rational.
• Theorem 3 (p. 4), with Remark 4 (pp. 4--5): the weighted series $\sum b_n/(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$ is irrational under polynomial bounds and $\limsup a_n^{1/c_{\mathbf w}^n}=\infty$.
• Theorem 5 (p. 5): for every $C>1$ a strictly increasing sequence with $\lim a_n^{1/\tilde c_{\mathbf w}^n}=C$ makes $\sum1/(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$ rational.

Read status. Claims checked: Theorems 2, 3 and 5 and Remarks 4 and 6 were read clause by clause on the arXiv v3 PDF; the proofs were read for structure only.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.