Problem 346

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}?

Source: erdosproblems.com/346

Number theory
Complete sequences

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

SOLVED, in the site's label (page last edited 1 September 2026). The site's remarks credit the counterexample that GPT Pro produced at Liam Price's prompting: a sequence with both deletion properties and $a_{n+1}/a_n\ge6/5$ whose ratios have the two subsequential limits $\varphi$ and $\varphi+1/4$ . The statement is disproved, so the answer to the question as posed is no. The derived standing, solved and disproved, is more specific than the site's label, which names no polarity: the accepted claim is a counterexample to the Statement. See the claim page (Price, 2026), which also carries the site's proof-claim entry of 2026-07-15.

Problem 346

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Proof claims

Discussion