Problem 267

Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\cdots$ be an infinite sequence with $n_{k+1}/n_k \geq c>1$ . Must

\sum_k\frac{1}{F_{n_k}}

be irrational?

Source: erdosproblems.com/267

Irrationality

Wiki page

Status

ClaimedProved

A full solution has been claimed but not yet accepted. The statement is true.

Claimed. The site labels the problem OPEN (page last edited 18 January 2026). The case $c\ge2$ is settled by Badea's 1993 corollary, the accepted partial claim on the Badea claim page (1993), and the site's commentary records the case $1<c<2$ as open. One pending full claim, a Lean 4 proof of 2026, asserts the answer yes for every $c>1$ ; the frontmatter standing follows from it.

An89: André-Jeannin, Richard, Irrationalité de la somme des inverses de certaines suites récurrentes. C. R. Acad. Sci. Paris Sér. I Math. (1989), 539-541.
Ba87: Badea, C., The irrationality of certain infinite series. Glasgow Math. J. (1987), 221-228.
Ba93: Badea, C., A theorem on irrationality of infinite series and applications. Acta Arith. (1993), 313-323.
BiHo76: Hoggatt, Jr., V. E. and Bicknell, Marjorie, A reciprocal series of Fibonacci numbers with subscripts $2^nk$ . Fibonacci Quart. (1976), 453-455.
Go74: Good, I. J., A reciprocal series of Fibonacci numbers. Fibonacci Quart. (1974), 346.

Problem 267

Status

Proof claims

Discussion