Problem 270

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claims/: The 2 claim pages of Problem 270, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(n)\to \infty$ as $n\to \infty$. Is it true that

$$\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}$$

is irrational?

Status. Disproved: Crmarić and Kovač's 2025 theorem that every positive real number is the value of such a series for some $f(n)\to\infty$ is recorded on its claim page. The site (page last edited 2025-09-28) labels the problem DISPROVED (LEAN) and credits them with the negative answer; the Lean proof behind the label is a public formalization of their argument linked from the claim page, not built or audited in this corpus. The variant with nondecreasing $f$, which the statement does not impose, remains open. The case $f(n)=n$ is answered yes: Kovač posted on the site's discussion thread (2026-07-12) that $\sum_n n!/(2n)!$ equals $\sum_n 1/(a_1\cdots a_n)$ with $a_k=4k-2$, a Cantor series with increasing integer terms, hence irrational. A manuscript of August 2026 extends the argument to $f(n)=an+b$ with $0\le b\le a$ and claims transcendence, hence irrationality, for every $a\ge1$ and $b\ge1-a$; it is a pending partial claim at Lambrinoudis 2026.

Source. erdosproblems.com/270, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #270, https://www.erdosproblems.com/270.

References.

• [CrKo25] T. Crmarić and V. Kovač, On the irrationality of certain super-polynomially decaying series. arXiv:2504.18712 (2025).
• [Ha75] Hansen, E. R., A Table of Series and Products. Prentice-Hall (1975), 87.

Formalization. Statement in formal-conjectures at the linked commit, tagged research solved, whose formal_proof attributes cite a Lean 4 proof in the public lean-proofs repository at a commit of 2026-09-15; that proof is linked from the claim page and is not built or audited in this corpus.

Progress

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Known Results

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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.

• crmaric_2025_irrationality_certain_super_polynomially_decaying_series
• crmaric_2025_irrationality_certain_super_polynomially_decaying_series / lemma_4
• crmaric_2025_irrationality_certain_super_polynomially_decaying_series / theorem_1
• crmaric_2025_irrationality_certain_super_polynomially_decaying_series / theorem_2