OPEN
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?
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].
