Erdős Problem #871

The problem

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 $2$ ?

Statement quoted from erdosproblems.com/871, maintained by Thomas Bloom. Original source: [ErNa88]. Current status there: disproved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date	Kind	AI systems	Outcome	Literature
6 Dec, 2025	Literature search	ChatGPT Deep research	🟡 Partial results found	—
5 Jan, 2026	AI building on known literature	Claude Opus 4.5, Gemini 3 Pro	🟢 Existing partial result upgraded to full solution (Lean)	🟡 Erdős and Nathanson (1989)

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

∎