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Updated Oct 8

Problem 871

claims/: The 1 claim page of Problem 871, one per claimant's result; the problem's standing derives from them.

Statement. Let $A$ be an additive basis of order $2$ , and suppose $1_A\ast 1_A(n)\to \infty$ as $n\to \infty$ . Can $A$ be partitioned into two disjoint additive bases of order $2$ ?

Status. DISPROVED (LEAN). The site credits the disproof to Larsen using Claude Opus 4.5; Larsen's thread posts describe a multi-agent system of Claude and Gemini agents (Claude 4.5 and Gemini 3 Pro in his 2026 preprint), which produced the Lean proof and its write-up, posted in January 2026: a small modification of the construction of [ErNa89] gives a basis of order two whose representation counts tend to infinity but which is not a union of two disjoint bases of order two. The label's Lean qualifier refers to Larsen's Lean proof as posted on the thread; a later revision of it is kept in Boris Alexeev's lean-proofs repository, and this corpus has built or audited neither. The accepted claim is Larsen.

Source. erdosproblems.com/871, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #871, https://www.erdosproblems.com/871.

References.

[ErNa88] Erdős, Paul and Nathanson, Melvyn B., Partitions of bases into disjoint unions of bases. J. Number Theory (1988), 1-9.
[ErNa89] Erdős, Paul and Nathanson, Melvyn B., Additive bases with many representations. Acta Arith. (1989), 399-406.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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Linked from (12)

Larsen's unsplittable basis with unbounded representation counts problems/additive_bases/E0871/claims Additive Bases and Sidon Sets additive_bases/erdos_1988_partitions_bases_into_disjoint_unions_bases Part 3, Problem 2 (p. 8): does f(n) -> infinity suffice to split an asymptotic basis of order 2 into two disjoint asymptotic bases of order 2 Theorem 3 (p. 5): an asymptotic basis of order 2 with at least c log n representations splits into t disjoint asymptotic bases of order 2 Theorem 4 (p. 6): an asymptotic basis of order h with at least c log n disjoint representations splits into t disjoint asymptotic bases of order h Theorem 5 (p. 6): an asymptotic basis of order h with f(n)/log n tending to infinity is a disjoint union of infinitely many asymptotic bases of order h additive_bases/erdos_1989_additive_bases_many_representations additive_bases/larsen_2026_three_questions_erdos_nathanson_asymptotic_bases Theorem 1 (p. 1): divergent representation function, splitting into two bases and containing a minimal basis are independent Theorem 2 (p. 2): two disjoint sets on the intervals between N_k = 4^{k+1} whose sums at N_{k+1} all meet a prescribed set

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