Problem 247

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claims/: The 1 claim page of Problem 247, one per claimant's result; the problem's standing derives from them.

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Statement. Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that

$$\limsup \frac{a_n}{n}=\infty.$$

Is

$$\sum_{n=1}^\infty \frac{1}{2^{a_n}}$$

transcendental?

Status. Open.

Source. erdosproblems.com/247, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #247, https://www.erdosproblems.com/247.

References.

• [Er75c] Erdős, P., Some problems and results on the irrationality of the sum of infinite series. J. Math. Sci. (1975), 1-7 (1976).
• [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. L., Old and new problems and results in combinatorial number theory. Monogr. Enseign. Math. 28 (1980), p. 61.
• [Er57] Erdős, P., On the irrationality of certain series. Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), 212--219, pp. 213, 215.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation (page last edited 20 January 2026) asks whether $\sum_{n\ge1}2^{-a_n}$ is transcendental for every increasing sequence of positive integers with $\limsup a_n/n=\infty$, and labels the problem OPEN. No claim answers that question. One accepted partial claim records the transcendence the site credits under the stronger condition $\limsup a_n/n^t=\infty$ for every $t\ge1$: Erdős 1957, Theorem 2 of [Er57] (printed p. 215), which shows that $\sum_k1/t^{n_k}$ satisfies no integer polynomial equation of degree at most $l$ when $\limsup n_k/k^l=\infty$; applied for every $l$ at $t=2$ it gives the credited statement, and under the problem's own hypothesis ($l=1$) it gives irrationality only. The result page Theorem 2 carries the statement and the structure of the proof.

The site's citation. The site credits the statement to [Er75c]. The sentence follows [ErGr80], p. 61, which reports the result as known for some time and cites its key [Er (75)], resolved by the monograph's bibliography (p. 112) to J. Math. Sci. 10 (1975), 1--7. That paper does not contain the statement. Its Theorem 2 (pp. 1--2) makes $\sum1/n_k$ a Liouville number when $n_k>k^{1+\varepsilon}$ and $\limsup n_k^{1/t^k}=\infty$ for every $t$; for $n_k=2^{a_k}$ this asks $\limsup a_k/t^k=\infty$ for every $t$, a class inside the one the 1957 theorem covers, so it gets no claim page. The open problem it states on p. 2 asks whether $\sum_kn_k/2^{n_k}$ is irrational when $\limsup n_k/k=\infty$, a different series. The card erdos_1976_problems_results_irrationality_sum_infinite_series digests the paper. The 1957 paper (p. 213) credits the transcendence under $\limsup\log n_k/\log k=\infty$, the same condition, to Erdős and Straus, Elem. Math. 9 (1954), p. 18, Problem 154; that item is a posed problem whose printed solution is not identified by the sources cited here, so it has no claim page of its own, and the 1957 theorem is the published proof.

Later statements. [Er88c], p. 106, restates the transcendence for sequences with $n_k/k^l\to\infty$ for every $l$ and asks whether $n_k>ck^2$ rules out a quadratic value; the site's remark rewrites this question, which the 1957 paper already asks on p. 213 for the square of the sum. The formal-conjectures file states the problem as erdos_247, tagged research open, and also a variant erdos_247.variants.strong_condition, the credited statement with the hypothesis $\limsup n_k/k^t=\infty$ for every real $t\ge1$, tagged research solved and citing [ErGr80], both with sorry proofs (the file at this revision); a statement file is not a formalization of a proof, and neither declaration is formalized evidence.

Dated search scope (2026-10-07). The site's page, its discussion thread and its proof-claims tab (none registered) record no proof claim and no result on the exact question beyond the statements above; no wider literature search was made. The site's label OPEN matches the derived standing: the one claim is partial.

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.

• erdos_1957_irrationality_certain_series
• erdos_1957_irrationality_certain_series / theorem_2
• erdos_1976_problems_results_irrationality_sum_infinite_series
• erdos_1976_problems_results_irrationality_sum_infinite_series / question_p2
• erdos_1976_problems_results_irrationality_sum_infinite_series / theorem_2
• erdos_1988_irrationality_certain_series_problems_results
• kaneko_2026_refinements_erdos_s_irrationality_criterion_certain
• laursen_2024_transcendence_certain_sequences_algebraic_numbers