Erdős Problem #267 - Discussion thread

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?

#267: [ErGr80,p.65]

irrationality

Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.

It may be sufficient to have $n_k/k\to \infty$. Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\sum \frac{1}{F_{2^n}}$ is irrational.

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

This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.

View the LaTeX source

This page was last edited 27 September 2025.

External data from the database - you can help update this
Formalised statement? Yes

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #267, https://www.erdosproblems.com/267, accessed 2025-11-16

0 comments on this problem

Order by oldest first or newest first.

All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic.

Back to the forum