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Updated Oct 7

Problem 264

claims/: The 2 claim pages of Problem 264, one per claimant's result; the problem's standing derives from them.

Statement. Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$ ) the sum

\sum \frac{1}{a_n+b_n}

is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?

Status. Open. The site labels the problem OPEN (page last edited 20 January 2026). Its two questions are the problem's two parts. The powers-of-two part is answered no by Kovač and Tao's accepted partial claim. Their Corollary 2.6 (Acta Math. Hungar. 175 (2025)) shows that no strictly increasing sequence with bounded successive ratios is an irrationality sequence of this type. An independent Lean proof by Aristotle is a pending partial claim of the same answer. The factorial part has no claim, and those results do not apply to factorials.

Source. erdosproblems.com/264, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #264, https://www.erdosproblems.com/264.

References.

[Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

Formalization. Statement in formal-conjectures.

Current assessment

The powers-of-two part is answered no by Kovač and Tao's Corollary 2.6, published in Acta Math. Hungar. 175 (2025) and recorded on its claim page; the corpus has checked the statement against the paper and has not independently reviewed the proof. The factorial part is open in the sources named on this page, the site record and the Kovač–Tao paper, and no literature search beyond them is recorded. Theorem 2.4's factorial-like denominators do not supply the bounded perturbations required by the question.

Known Results

Kovač and Tao's Corollary 2.6 answers the powers-of-two case negatively: there is a bounded nonzero integer perturbation for which the reciprocal series is rational. The result is recorded on their accepted partial claim page, and an independent Lean proof of the same answer by Aristotle, posted by Boris Alexeev, on its own pending claim page. Their Theorem 2.4 allows a rational sum with denominators asymptotic to $n!$ , but the perturbations it constructs are not bounded, so it does not answer the factorial case. See Kovač–Tao.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (8)

problems/irrationality/E0264/claims Irrationality and Diophantine Approximation Borwein: On the irrationality of Σ (1/(q^n+r))Theorem 4 (p. 257): the sum of 1/(q^n-r) is irrational Borwein: On the irrationality of certain series Theorem 1 (p. 142): the sum of 1/(q^n+c) is irrational irrationality/erdos_1988_irrationality_certain_series_problems_results irrationality/kovac_2024_several_irrationality_problems_ahmes_series

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