Problem 869

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claims/: The 1 claim page of Problem 869, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $n\not\in A+A$)?

Status. 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.

Source. erdosproblems.com/869, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #869, https://www.erdosproblems.com/869.

References.

• [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: erdos_1992_my_forgotten_problems_number_theory.
• [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.

Formalization. Statement in formal-conjectures. A Lean 4 formalization of the construction, produced with Codex and GPT-5.6 Sol and posted on 2026-08-17 in lean-proofs, states the negative answer; this corpus has not audited its statement.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

• erdos_1988_partitions_bases_into_disjoint_unions_bases
• erdos_1988_partitions_bases_into_disjoint_unions_bases / problem_4
• larsen_2026_three_questions_erdos_nathanson_asymptotic_bases
• larsen_2026_three_questions_erdos_nathanson_asymptotic_bases / theorem_1
• larsen_2026_three_questions_erdos_nathanson_asymptotic_bases / theorem_2