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OP-8914OpenMathematics · general

Erdős #267 · irrationality

Let F1=F2=1F_1=F_2=1 and Fn+1=Fn+Fn−1F_{n+1}=F_n+F_{n-1} be the Fibonacci sequence. Let n1<n2<⋯n_1<n_2<\cdots be an infinite sequence with nk+1/nk≥c>1n_{k+1}/n_k \geq c>1. Must

∑k1Fnk\sum_k\frac{1}{F_{n_k}}

be irrational?

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let F1=F2=1F_1=F_2=1 and Fn+1=Fn+Fn−1F_{n+1}=F_n+F_{n-1} be the Fibonacci sequence. Let n1<n2<⋯n_1<n_2<\cdots be an infinite sequence with nk+1/nk≥c>1n_{k+1}/n_k \geq c>1. Must

∑k1Fnk\sum_k\frac{1}{F_{n_k}}

be irrational?

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Background

It may be sufficient to have nk/k→∞n_k/k\to \infty. Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that ∑1F2n\sum \frac{1}{F_{2^n}} is irrational - in fact,

∑1F2n=7−52.\sum \frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}.

Badea [Ba87] proved that ∑1F2n+1\sum \frac{1}{F_{2^n+1}} is irrational.

The sum ∑1Fn\sum \frac{1}{F_n} itself was proved to be irrational by André-Jeannin [An89].

The main problem has been proved for c≥2c\geq 2 by Badea [Ba93]. It remains open for 1<c<21<c<2.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #267 link
  2. [2]Upstream Lean 4 statement for Erdős #267 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 267 link

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