---

Claim. Let $n(1)<n(2)<\cdots$ be positive integers with $n(k+1)\ge2n(k)-1$ for all sufficiently large $k$. Then $\sum_{k\ge1}1/F_{n(k)}$ is irrational. In particular the answer to Problem 267 is yes whenever the ratio constant satisfies $c\ge2$, since $n_{k+1}/n_k\ge c\ge2$ gives $n_{k+1}\ge2n_k>2n_k-1$. The source is Badea, C., A theorem on irrationality of infinite series and applications, Acta Arith. 63 (1993), 313–323, on the card badea_1993_theorem_irrationality_infinite_series_applications. Its Corollary 3.2 states the result for the sequence $x_0=0$, $x_1=1$, $x_{n+2}=ax_{n+1}+bx_n$ with positive integers $a,b$, which is the Fibonacci sequence in the problem's indexing when $a=b=1$; the proof uses the identity $x_{2n+1}=x_{n+1}^2+bx_n^2$ of Lemma 3.1(i), which gives $x_{2n-1}\ge x_n^2$ and so $x_{n(k+1)}\ge x_{n(k)}^2>x_{n(k)}^2-x_{n(k)}+1$ for large $k$, and concludes by the paper's Corollary 2.2, which generalizes Theorem A, the author's 1987 criterion that $\sum b_n/a_n$ is irrational when $a_{n+1}>(b_{n+1}/b_n)a_n^2-(b_{n+1}/b_n)a_n+1$ for all large $n$. Badea writes after the corollary that it gives an affirmative answer to Problem B, the problem of Erdős and Graham recorded here, in the case $c\ge2$, and notes in Section 3 that André-Jeannin's irrationality of $\sum1/F_n$ indicates that the answer may be affirmative for $1<c<2$ as well.

Covers. Every ratio constant $c\ge2$, and more generally every index sequence with $n(k+1)\ge2n(k)-1$ for all large $k$. That condition contains the problem's recorded instances, each with its own accepted partial claim: $n_k=2^k$ (Good 1974 on the Good page and Hoggatt and Bicknell 1976 on the Hoggatt–Bicknell page, who evaluate the sum as $(7-\sqrt5)/2$), $n_k=2^k+1$ (Badea 1987 on the Badea 1987 page, with equality in the condition) and $n_k=n^k$ for an integer $n\ge2$ (cited in the site's thread on 2026-04-30). Not covered: the range $1<c<2$, which the site's commentary records as open and which the pending full claim on the Snyder page asserts.

Acceptance. Refereed: Acta Arithmetica, volume 63, issue 4 (1993). The site's curator writes that the main problem has been proved for $c\ge2$ by Badea [Ba93] but labels the problem OPEN, so that commentary is not listed as reviewed evidence. The corpus has not reproved the theorem and awards no tier of its own.

Depends on. Nothing in this wiki; the claim rests on the cited paper.