OPEN
This is open, and cannot be resolved with a finite computation.
Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that
- $A\backslash B$ is complete for any finite subset $B$ and
- $A\backslash B$ is not complete for any infinite subset $B$.
(Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of the sequence.)
Is it true that if $a_{n+1}/a_n \geq 1+\epsilon$ for some $\epsilon>0$ and all $n$ then\[\lim_n \frac{a_{n+1}}{a_n}=\frac{1+\sqrt{5}}{2}?\]
Graham
[Gr64d] has shown that the sequence $a_n=F_n-(-1)^{n}$, where $F_n$ is the $n$th Fibonacci number, has these properties. Erdős and Graham
[ErGr80] remark that it is easy to see that if $a_{n+1}/a_n>\frac{1+\sqrt{5}}{2}$ then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties.
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This page was last edited 07 December 2025.
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T. F. Bloom, Erdős Problem #346, https://www.erdosproblems.com/346, accessed 2026-01-16
