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Updated Oct 7

Problem 257

claims/: The 5 claim pages of Problem 257, one per claimant's result; the problem's standing derives from them.

Statement. Let $A\subseteq \mathbb{N}$ be an infinite set. Is

\sum_{n\in A}\frac{1}{2^n-1}

irrational?

Status. Open, the site's label (OPEN; page last edited 2026-04-15). No claim settles the question for every infinite support, so the frontmatter standing, open/none, follows. Accepted partial claim pages record the settled classes of supports: Erdős's 1948 theorem for $A=\mathbb N$ and its sets of multiples, Erdős's 1968 theorem for pairwise coprime supports with convergent reciprocal sum, Duverney and Tachiya's theorem for the sets $F_s(E)$ of products of powers below $s$ of a pairwise coprime, polynomially bounded sequence, such as the squarefree integers, and Tao and Teräväinen's theorem for the primes. One pending partial claim, Cook's 2026 note, asserts irrationality for every support with convergent reciprocal sum.

Source. erdosproblems.com/257, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #257, https://www.erdosproblems.com/257.

References.

[Er48] Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.
[Er68d] Erdős, P., On the irrationality of certain series. Math. Student (1968), 222-226.
[Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.
[KoTa24] Kovač, V. and Tao, T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024); Acta Math. Hungar. 175 (2025), 572–608.
[TaTe25] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers. arXiv:2512.01739 (2025); version 2 of 25 April 2026; library card: tao_2025_quantitative_correlations_problems_prime_factors_consecutive.

Formalization. Statement in formal-conjectures.

Current assessment

The question for arbitrary infinite supports is open: the site labels Problem 257 OPEN (page last edited 2026-04-15), and no result or claim settles it. The settled classes of supports, each with its claim page or its recorded reason, are these. $A=\mathbb N$ and its sets of multiples $d\mathbb N$ : Erdős's 1948 theorem at the bases $2$ and $2^d$ , the accepted partial claim on its claim page. Every pairwise coprime $A$ with $\sum_{a\in A}1/a<\infty$ , at every integer base: Erdős's 1968 theorem, accepted on its claim page. The sets $F_s(E)$ of products $\prod e_i^{m_i}$ with $0\le m_i<s$ over a pairwise coprime, polynomially bounded sequence $E$ , the squarefree integers and the integers coprime to a fixed modulus among them: Duverney and Tachiya's Corollary 1.2 of 2019, accepted on its claim page. The primes: Theorem 1.3 of Tao and Teräväinen's preprint, which the site's curator accepts through Problem 69 and which is accepted on its claim page; the paper only sketches the prime powers. Every $A$ with $\sum_{a\in A}1/a<\infty$ , at every integer base: Theorem 1.1 of Cook's note of 2026, drafted with AI agents and without independent review, the pending partial claim on its claim page. Two further classes have no claim page because their source states no instance of the problem. Borwein's Theorem 1 (Math. Proc. Cambridge Philos. Soc. 112 (1992), 141--146; card) proves $\sum_{n\ge1}1/(q^n+r)$ irrational for every integer $|q|>1$ and nonzero rational $r\ne-q^n$ ; the card's specialization, $q=2^d$ and $r=-2^{-a}$ , gives every single arithmetic progression $\{a+dk:k\ge0\}$ and every cofinite set. Tachiya's Theorem 1 (Tokyo J. Math. 27 (2004), no. 1, DOI 10.3836/tjm/1244208475), raised in the site's thread on 2025-09-05, proves $\sum_{n\ge1}a_n/(1-q^n)$ irrational for every integer $q\ge2$ and every period-two integer sequence $a_n$ not identically zero; the thread's specialization gives the even and the odd integers, which the thread notes already follow from Erdős's and Borwein's theorems. The formal-conjectures catalog has tagged its variant for $A=\mathbb N$ , erdos_257.variants.tsum_top, research solved with a formal proof link since 2026-09-23, while its main statement stays research open; the link is recorded on the 1948 claim page.

A variant the site discusses settles no instance of the problem. Erdős speculated in 1988 that $\sum_{n\in A}1/(2^n-t_n)$ is irrational for every infinite $A$ and every bounded integer sequence $t_n$ . This is false: Kovač and Tao (Acta Math. Hungar. 175 (2025), 572--608, Theorem 2.5; card) disprove it for nonzero $|t_n|<C$ already at $A=\mathbb N$ , and Kovač's thread comment of 2025-10-30 sketches a choice with $1\le t_n\le6$ over $A=\{n\ge100\}$ for which the sum is rational. Dated search scope: the site's page and remarks (2026-09-04), its discussion thread (posts through 2026-09-11), the arXiv record of Tao and Teräväinen's preprint (2026-09-06) and the formal-conjectures file at its commit of 2026-09-23; no wider literature search is recorded, and no proof is checked here.

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 (28)

Problem 69 Tao and Teräväinen prove the prime-factor series irrational Problem 250 Erdős's irrationality of the full Lambert series at every integer base Erdős's irrationality theorem for pairwise coprime supports Duverney and Tachiya's Lambert series over products of coprime generators Tao and Teräväinen's theorem for the primes Cook's note on supports with convergent reciprocal sum problems/irrationality/E0257/claims Irrationality and Diophantine Approximation arithmetic_functions/tao_2025_quantitative_correlations_problems_prime_factors_consecutive Irrationality and Diophantine Approximation Borwein: On the irrationality of certain series Theorem 1 (p. 142): the sum of 1/(q^n+c) is irrational Theorem 2 (p. 145): the alternating sum of 1/(q^n+c) is irrational Duverney–Tachiya: Refinement of the Chowla–Erdős method and linear independence of certain Lambert series Corollary 1.2 (p. 4): 1 and the Lambert series over F_s(E) of 1/(q^{jn^i} - 1) are linearly independent Lemma 4.1 (p. 9): a divisor-function bound on theta gives the progression condition (H_2)Theorem 1.1 (pp. 2-3): a rational series sum theta(n)/q^n forces zeros of theta in every progression Theorem 1.2 (p. 3): a linear relation among the series sum theta_i(n)/q^{jn} forces zeros of a combination irrationality/erdos_1948_arithmetical_properties_lambert_series irrationality/erdos_1969_irrationality_certain_series Theorem (p. 222, unnumbered): irrationality for pairwise coprime exponents irrationality/erdos_1988_irrationality_certain_series_problems_results irrationality/kovac_2024_several_irrationality_problems_ahmes_series irrationality/postelmans_2007_irrationality_zeta_q_1_zeta_q_2 Theorem 1.1 (p. 2): integer approximations to zeta_q(1) with error decaying like p^(-c n^2)Theorem 1.3: 1, zeta_q(1) and zeta_q(2) are linearly independent over Q

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