Erdős Problem #267 — 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?

Erdős and Graham posed the question in $1980$: whether every sufficiently lacunary reciprocal Fibonacci sum is irrational.

Known results

• Badea proved irrationality for $c\geq 2$ in $1993$.
• Good ($1974$) and Bicknell–Hoggatt ($1976$) found $\sum_{k\geq0}1/F_{2^k}=(7-\sqrt{5})/2$.
• André-Jeannin proved $\sum 1/F_n$ irrational in $1989$.
• Transcendence holds for arbitrary sequences when $c>2$; this strengthens earlier bounds.

AlphaProof claim

A formalization comment says “Formal proof provided by AlphaProof,” but the theorem remains marked open and contains sorry; no completed proof or independent mathematical artifact is available.

Current status (as of March 2026): Irrationality is settled for $c\geq2$ and transcendence for $c>2$, while the original range $1<c<2$ remains open; the AlphaProof solution claim is unverified.