Problem 265

../

claims/: The 3 claim pages of Problem 265, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $1\leq a_1<a_2<\cdots$ be an increasing sequence of integers. How fast can $a_n\to \infty$ grow if

$$\sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n-1}$$

are both rational?

Formulation. The site notes that the original source is ambiguous as to what the problem is, and the community database marks the problem with an ambiguous statement. The hypothesis that the sequence be strictly increasing was added to the site's statement after Vjekoslav Kovač's comment of 2026-01-19 (post), which notes that without it the remaining question would be trivial, since any sequence making both sums rational could be rearranged to grow as fast as one likes along a subsequence.

Status. Open. The site labels the problem OPEN (page last edited 21 January 2026). Its commentary says that Kovač and Tao [KoTa24] have almost completely solved the problem by constructing a sequence growing doubly exponentially, $a_n^{1/\beta^n}\to\infty$ for some $\beta>1$, and that the remaining question is the exact exponent, in particular whether $\limsup a_n^{1/2^n}>1$ is possible, since a folklore result makes the sum irrational once $\lim a_n^{1/2^n}=\infty$. Their result is the accepted partial claim on [[problems/irrationality/E0265/claims/2024_11_27_kovac_tao|the Kovač–Tao claim page]], every base $\beta<\sqrt{6/5}$. Two pending partial claims follow: [[problems/irrationality/E0265/claims/2026_08_28_cam|a residual-state construction of 2026]] asserts that every exponent $\beta<(\sqrt{13}-1)/2$ can be reached, and [[problems/irrationality/E0265/claims/2026_09_07_kitamura|a Lean 4 development of 2026]] asserts that $\limsup a_n^{1/2^n}>1$ is impossible. The problem asks for the exact growth threshold, which no claim determines.

Source. erdosproblems.com/265, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #265, https://www.erdosproblems.com/265.

References.

• [KoTa24] Kova\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

Formalization. No statement in formal-conjectures (no file for the problem). A Lean 4 development posted on 2026-09-07, claiming that no such sequence has $\limsup a_n^{1/2^n}>1$, is linked from its claim page; the corpus records no build or audit of it.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

• kovac_2024_several_irrationality_problems_ahmes_series