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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 2 is stated on pp. 2--3 of that PDF. Bibliographic details and the edition read are on the source card.

Statement

Fix a positive integer $d$ and let $\psi>1$ be the unique positive solution of $\psi^d=\psi^{d-1}+1$.

Part (1) (p. 3). If $\{a_n\}_{n=1}^\infty$ is a monotonically increasing sequence of positive integers (the paper's footnote 2: non-decreasing) with

$$\lim_{n\to\infty}a_n^{1/\psi^n}=\infty,$$

then the sum

$$\sum_{n=1}^{\infty}\frac{1}{a_na_{n+1}\cdots a_{n+d-1}}$$

(the paper's (2.4)) is irrational.

Part (2) (p. 3). For every $C\in(1,\infty)$ there is a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ with $\lim_{n\to\infty}a_n^{1/\psi^n}=C$ for which the sum (2.4) is rational.

The case $d=2$ (p. 3). Then $\psi$ is the golden ratio $\phi=(1+\sqrt5)/2<2$. The paper draws two consequences: the hypothesis $\liminf_{n\to\infty}a_n^{1/2^n}>1$ of its Question 1 suffices for $\sum_{n\ge1}1/(a_na_{n+1})$ to be irrational, while the hypothesis $\liminf_{n\to\infty}a_n^{1/\phi^n}>1$ does not.

Proof pointer

The paper does not prove Theorem 2 separately (p. 3). For $d=1$, where $\psi=2$, it says Part (1) is an easy consequence of Erdős's theorem (J. Math. Sci. 10 (1975), Theorem 1) and that the Sylvester sequence gives an explicit example for Part (2). For $d\ge2$ it says Part (1) is a special case of Theorem 3 and Part (2) a particular instance of Theorem 5. With all weights equal to $1$ the defining polynomials of $c_{\mathbf w}$ and $\tilde c_{\mathbf w}$ both reduce to $x^d-x^{d-1}-1$. The heuristic on p. 3, which the paper says Tao sketched for the golden ratio, compares the denominator of the $N$-th partial sum with the size of the tail for $a_n\approx\exp(c^n)$ and arrives at $c^d-1\le c^{d-1}$, that is $c\le\psi$.

Read depth. Claims checked: the statement, footnote 2 and the paragraph after the theorem were read clause by clause on pp. 2--3 of the arXiv v3 PDF. The reduction to Theorems 3 and 5 is the paper's own remark and was not re-derived here.

Dependencies

Theorem 3 and Theorem 5 for $d\ge2$; Erdős's 1975 theorem on $\sum1/a_n$ and the Sylvester sequence for $d=1$.

Bears on

• Problem 1051: the paper's Question 1 quotes the problem, and the paper says (p. 3) that Part (1) with $d=2$ implies its hypothesis $\liminf a_n^{1/2^n}>1$ suffices for the irrationality of $\sum1/(a_na_{n+1})$, an affirmative answer. By the same remark, Part (2) with $d=2$ shows that the hypothesis $\liminf a_n^{1/\phi^n}>1$ does not suffice. The paper offers Theorem 2 (p. 2) as its answer to Erdős and Graham's request for the strongest theorem of this type.