Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\sum_n \frac 1 {F_{2^n}}$ is irrational.
Ref:
* [Go74] Good, I. J., _A reciprocal series of Fibonacci numbers_
* [BiHo76] Hoggatt, Jr., V. E. and Bicknell, Marjorie, _A reciprocal series of Fibonacci numbers with subscripts $2\sp{n}k$_
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