Problem 346

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claims/: The 2 claim pages of Problem 346, one per claimant's result; the problem's standing derives from them.

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Statement. 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

$$\lim_n \frac{a_{n+1}}{a_n}=\frac{1+\sqrt{5}}{2}?$$

Formulation. The site's wording follows Erdős and Graham ([ErGr80], p. 57), who ask whether every sequence with $s_{n+1}/s_n\ge1+\epsilon$ and both deletion properties must have $\lim_n s_{n+1}/s_n=(1+\sqrt5)/2$. Convergence of the ratios is part of the conclusion, and Price's counterexample refutes the Statement in that form. Erdős and Graham add that very irregular sequences, with $\liminf s_{n+1}/s_n=1$ and $\limsup s_{n+1}/s_n=\infty$, have both deletion properties; the ratio bound $1+\epsilon$ excludes them.

A commenter in the thread suggested, and Nat Sothanaphan agreed, that the question likely assumes the limit exists. Graham's own question in [Gr64d] (p. 10) is a different one: whether some sequence with both deletion properties is essentially different from $F_n-(-1)^n$, for example with $\lim t_{n+1}/t_n\ne\varphi$. No statement of the problem by Erdős or Graham assumes that the limit exists, so that reading is a variant. For it Kenta Kitamura posted on 2026-06-21 a Lean development, prepared with Codex and ChatGPT, whose theorems state that an existing limit $L>1$ must be $\varphi$. Sothanaphan checked it in the thread and noted that the case $L>\varphi$ also follows from Burr and Erdős (1981). Against the Statement it is the claimed partial claim Kitamura's limit-exists theorem, covering the sequences whose ratios converge; the answer yes on the variant is claimed, not accepted.

Status. 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, which also carries the site's proof-claim entry of 2026-07-15.

Source. erdosproblems.com/346, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #346, https://www.erdosproblems.com/346.

References.

• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
• [Gr64d] Graham, R. L., A property of Fibonacci numbers. Fibonacci Quart. (1964), 1-10.

Formalization. Statement in formal-conjectures. The counterexample's Lean proof, completed by GPT-5.5 in Codex, was posted as code embedded in a live.lean-lang.org link and is ported in the repository plby/lean-proofs, pinned on Price's claim page; Kitamura's Lean development for the limit-exists reading is pinned on his claim page. Nothing was built or audited here.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• graham_1964_property_fibonacci_numbers
• graham_1964_property_fibonacci_numbers / problem_p10
• graham_1964_property_fibonacci_numbers / theorem
• price_2026_counterexample_erdos_problem_346
• price_2026_counterexample_erdos_problem_346 / lemma_2
• price_2026_counterexample_erdos_problem_346 / theorem_1