How fast can $a_n$ grow if $\sum 1/a_n$ and $\sum 1/(a_n-1)$ are both rational? (Erdős #265)
Statement
Let $1\le a_1<a_2<\cdots$ be an increasing sequence of integers for which both $$\sum_n \frac{1}{a_n} \quad\text{and}\quad \sum_n \frac{1}{a_n-1}$$ are rational. How fast can $a_n\to\infty$ grow? In particular, decide whether $\limsup a_n^{1/2^n}>1$ is attainable for such a sequence.
Acceptance. FULLY RESOLVES: determine the exact attainable growth rate — resolve whether there exists an increasing integer sequence with both $\sum 1/a_n$ and $\sum 1/(a_n-1)$ rational and $\limsup a_n^{1/2^n}>1$: either an explicit such construction, with both rational sum values certified, or a proof that every valid sequence has $\limsup a_n^{1/2^n}\le 1$. A full written proof (or machine-checkable Lean proof) is required. ADVANCES: strictly improve the achievable growth exponent beyond the '$a_n^{1/\beta^n}\to\infty$ for some $\beta>1$' record stated in the background — exhibit a valid sequence with an explicitly larger $\beta$, giving the construction and certifying both rational sums; or tighten the folklore upper barrier below its current form. Deliver the construction plus rationality proofs, or the growth-bound proof.
Background
Posed by Erdős and Graham [ErGr80, p.64] and reiterated by Erdős in [Er88c, p.104]; listed as open on erdosproblems.com/265 (fetched 2026-07-13, status 'open', tagged 'irrationality'). Cantor observed that $a_n=\binom{n}{2}$ works, since both series then telescope to rationals. Replacing the shift $-1$ by another constant admits higher-degree polynomial examples — e.g. for $\sum_{n\ge 2}1/a_n$ and $\sum_{n\ge 2}1/(a_n-12)$ the sequence $a_n=n^3+6n^2+5n$ makes both series rational. Erdős believed $a_n^{1/n}\to\infty$ is achievable but that $a_n^{1/2^n}\to 1$ is necessary. Kovač and Tao [KoTa24] almost completely settled this, constructing such a sequence that grows doubly exponentially: there exists a valid sequence with $a_n^{1/\beta^n}\to\infty$ for some $\beta>1$. On the upper side, a folklore result gives that $\sum 1/a_n$ is irrational whenever $\lim a_n^{1/2^n}=\infty$, so no valid sequence can grow faster than base-2 double-exponential. The remaining gap is the precise exponent: whether $\limsup a_n^{1/2^n}>1$ is attainable, i.e. whether $\beta$ can be pushed up to the folklore barrier at base 2. Attacker's tool: explicit constructions of telescoping / rational reciprocal sums (partial-fraction and difference-equation families) combined with Diophantine irrationality bounds; formalisation to certify.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #265 (T. F. Bloom) | website |
Investigations · 0
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