Problem 869

If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\cup A_2$ contain a minimal additive basis of order $2$ (one such that deleting any element creates infinitely many $n\not\in A+A$ )?

Source: erdosproblems.com/869

Number theory
Additive bases

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SolvedDisproved

An accepted solution exists. The statement is false.

Disproved. The site records Larsen's construction of a union of two disjoint bases of order $2$ containing no minimal basis, one of eight cases showing that divergent representation counts, splitting into two bases, and containing a minimal basis are independent; the accepted claim is Larsen. A write-up posted to the problem's forum by Przemek Chojecki on 2026-04-25, in which GPT-5.5 Pro streamlines Larsen's construction, restates that result and is disclosed on the claim page rather than given its own.

ErNa88: Erdős, Paul and Nathanson, Melvyn B., Partitions of bases into disjoint unions of bases. J. Number Theory (1988), 1-9.
Ha56: Härtter, Erich, Ein Beitrag zur Theorie der Minimalbasen. J. Reine Angew. Math. (1956), 170-204.
Na74: Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324-333.
Er92c: Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. 15 (1992), 34-50; p. 44 restates the question. Library home: Erdős 1992.
La26: Larsen, Daniel, Three questions of Erdős–Nathanson on asymptotic bases of order 2. arXiv:2603.03472 (2026), 7 pp.; note posted to the problem's forum 2026-01-25.

Problem 869

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