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Source. K. Barreto, J. Kang, S. Kim, V. Kovač and S. Zhang, Irrationality of rapidly converging series: a problem of Erdős and Graham, arXiv:2601.21442v3 (8 July 2026). Theorem 5, Remark 6 and Example 7 are on p. 5 of that PDF and the proof is Section 4 (pp. 12--14). Bibliographic details and the edition read are on the source card.

Statement

Theorem 5 (p. 5). Fix a positive integer $d$ and non-negative integers $\mathbf w=(w_0,w_1,\ldots,w_{d-1})$ with $w_{d-1}\ge1$, and let $\tilde c_{\mathbf w}$ be the largest positive root of

$$\widetilde P_{\mathbf w}(x)=(x-1)\sum_{j=0}^{d-1}w_jx^j-x^{d-1}.$$

Then for every $C\in(1,\infty)$ there is a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ with

$$\lim_{n\to\infty}a_n^{1/\tilde c_{\mathbf w}^n}=C$$

(the paper's (2.6)) and

$$\sum_{n=1}^{\infty}\frac{1}{a_n^{w_0}a_{n+1}^{w_1}\cdots a_{n+d-1}^{w_{d-1}}}\in\mathbb Q.$$

Remark 6 (p. 5). The largest real root of $\widetilde P_{\mathbf w}$ lies in $(1,\infty)$ because $\widetilde P_{\mathbf w}(1)<0$. When $w_0,\ldots,w_{d-1}\in\{0,1\}$, Theorem 5 shows that Theorem 3 is sharp; there $W=1$ and $\widetilde P_{\mathbf w}=P_{\mathbf w}$, so $\tilde c_{\mathbf w}=c_{\mathbf w}$ (Section 5, p. 15). For other weights the two roots may differ, and the paper leaves open which is nearer the true threshold (p. 15).

Example 7 (p. 5). For $\mathbf w=(1,0,2,1)$ the paper computes $c_{\mathbf w}=1.914\ldots$ and $\tilde c_{\mathbf w}=1.345\ldots$, so Theorems 3 and 5 leave a gap for $\sum1/(a_na_{n+2}^2a_{n+3})$.

Proof pointer

Section 4, pp. 12--14, by an interval-filling argument. Lemma 14 (p. 13): if $\beta_n\le\gamma_n$ are monotonically increasing positive-integer sequences with the series at $\beta$ convergent, $\beta_n/\gamma_n\to0$, and a further growth condition (4.3) linking consecutive terms, then the set of sums of the series over all choices $a_n\in[\beta_n,\gamma_n]\cap\mathbb N$ is a finite union of non-degenerate closed bounded intervals, and so contains a rational number. The proof of Theorem 5 (p. 14) takes $\beta_n=\lfloor C^{c^n+n^2+1}\rfloor$ and $\gamma_n=\lfloor C^{c^n+n^2+n}\rfloor$ with $c=\tilde c_{\mathbf w}$, checks (4.3) using that $c$ is a root of $\widetilde P_{\mathbf w}$, and replaces finitely many terms by $a_n=n$ to make the sequence strictly increasing, which does not affect rationality.

Read depth. Claims checked: Theorem 5, Remark 6 and Example 7 were read clause by clause on p. 5 of the arXiv v3 PDF, and Section 5 on p. 15; the proof was read for structure only.

Dependencies

None in the corpus; the construction is self-contained (Lemma 14).

Bears on

• Problem 1051: the paper says (p. 3) that Part (2) of Theorem 2 for $d\ge2$ is a particular instance of Theorem 5; for $d=2$ and weights $(1,1)$, so $\tilde c_{\mathbf w}=\phi$, it gives strictly increasing sequences of positive integers with $\lim a_n^{1/\phi^n}=C$ for every $C>1$ and a rational $\sum1/(a_na_{n+1})$. This bears on how far the problem's growth hypothesis can be weakened, not on its answer.