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Source. C. Badea, The irrationality of certain infinite series, Glasgow Math. J. 29 (1987), no. 2, 221--228, doi:10.1017/S0017089500006868. Corollary 4 and its proof are on p. 227 (Section 5). Bibliographic details are on the source card.

Statement

The Fibonacci numbers are $F_0=0$, $F_1=1$, $F_{n+2}=F_{n+1}+F_n$ (p. 226).

Corollary 4 (p. 227, quoted). "The sum of the series $\sum_{n=1}^{\infty}1/F_{2^n+1}$ is an irrational number."

Section 5 (pp. 226--227) presents this as the answer to the first of two questions Erdős and Graham raise in Old and new problems and results in combinatorial number theory (1980), pp. 64--65, where they write that nothing is known about the character of this sum.

Proof pointer

P. 227. By Corollary 1 it suffices that $F_{2^{n+1}+1}\ge F_{2^n+1}^2$ for all large $n$, which follows from the identity $F_{2k+1}=F_k^2+F_{k+1}^2$ at $k=2^n$.

Read depth. Claims checked: the statement was read on p. 227 of the print and the short proof was followed.

Dependencies

Corollary 1 and the identity $F_{2k+1}=F_k^2+F_{k+1}^2$.

Bears on

• Problem 267: the index sequence $n_k=2^k+1$ has $n_{k+1}/n_k\ge5/3$ for $k\ge1$, so the corollary answers the problem yes for that single sequence. It says nothing about other index sequences.