OPEN
This is open, and cannot be resolved with a finite computation.
Let $A=\{a_1<\cdots<a_k\}$ be a finite set of 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?
An old problem of Dickson. Even a starting set as small as $\{1,4,9,16,25\}$ requires thousands of terms before periodicity occurs.
This problem is discussed under Problem 7 on
Green's open problems list.
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Additional thanks to: Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #341, https://www.erdosproblems.com/341, accessed 2026-01-14
