Problem 341

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?

Source: erdosproblems.com/341

Number theory

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SolvedDisproved

An accepted solution exists. The statement is false.

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.

Problem 341

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