Problem 341

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claims/: The 1 claim page of Problem 341, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A=\{a_1<\cdots<a_k\}$ be a finite set of positive integers and extend it to an infinite sequence $\overline{A}=\{a_1<a_2<\cdots \}$ by defining $a_{n+1}$ for $n\geq k$ to be the least integer exceeding $a_n$ which is not of the form $a_i+a_j$ with $i,j\leq n$. Is it true that the sequence of differences $a_{m+1}-a_m$ is eventually periodic?

Status. Disproved; the site's label is OPEN (on 2026-10-07; page last edited 20 January 2026). The corpus accepts Li's full disproof of 9 August 2026, an explicit 21-element seed whose greedy extension has aperiodic gaps, on formalized evidence: Boris Alexeev's Lean formalization of it was built here and its theorem audited against the Statement above, so the problem stands solved and disproved. The site has not accepted the claim, and no refereed version or independent review was found.

Source. erdosproblems.com/341, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #341, https://www.erdosproblems.com/341.

Formalization. Statement in formal-conjectures, tagged research solved on 2026-09-18 with a formal_proof link to Boris Alexeev's formalization of Li's result. This corpus built that formalization at a pinned commit and audited its theorem against the Statement; Li's claim page records the build and links the statement and the formalization at their pins.

Current assessment

The frontmatter standing is derived from the one claim page, Li's full claim that the answer is no, accepted on formalized evidence, so the problem stands solved and disproved while the site's label is OPEN. The audited Lean theorem asserts that some finite seed has a greedy extension whose gaps are not eventually periodic; it does not name Li's seed, though its proof uses it, and Li's infinite family of seeds is not built. The notes under Known Results record outside results and claims and are not independently reviewed. Search scope, 2026-10-05 and 2026-10-07: the site's page, thread and proof-claims tab, the claimant's repository and its Zenodo record, the formal-conjectures catalog and the community database; no literature search beyond these is recorded.

Known Results

Accepted on formalized evidence. Li's paper Counterexamples to Erdős Problem 341 (9 August 2026; the claim page) exhibits the seed $\{1,2,3,5,7,13,22,27,28,32,36,40,47,48,52,63,71,77,81,89,97\}$ and proves that the gaps of its greedy extension, with equal summands allowed as the rule above allows, are not eventually periodic: an explicit infinite set built from a scale-eight controller inside three residue classes modulo $49$ satisfies the exact greedy recurrence beyond $97$, and the controller's aperiodicity transfers to the gaps; a modification gives an infinite family of seeds. The accompanying Lean 4 development states the fixed-seed theorem and the family's, and its README reports the standard axioms only; that repository is not built here. Boris Alexeev's formalization of the result in his public repository carries Li's fixed-seed files; this corpus built it at a pinned commit, checked its axioms and its fingerprint against the comparator challenge, and audited its theorem, as the claim page records. The formal-conjectures collection tagged the catalog statement research solved on 2026-09-18, crediting Li and pointing its formal_proof attribute at Alexeev's file (on 2026-10-05 the site showed OPEN and the community database said open). The site's discussion thread holds one comment (8 July 2026) reporting that the example $\{1,4,9,16,25\}$ of the site's commentary becomes periodic with period $224$ from its 87th term under the rule above, with a computation linked; that comment concerns the commentary's example, not the question.