Erdős Problem #871

Partial progress by Claude Opus 4.5 and Gemini 3 Pro (5 Jan 2026): existing partial result upgraded to full solution (Lean).

Problem
Erdős problem #871 (number theory, additive basis); current status: disproved (Lean)
Credit
Claude Opus 4.5 and Gemini 3 Pro · AI building on known literature
Checked
Lean formalizationListed in Tao's wiki
Timeline
First AI contribution 2025-12-06 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
PARTIALPer Tao's AI-contributions wiki

The problem

Let AA be an additive basis of order 22, and suppose 1A∗1A(n)→∞1_A\ast 1_A(n)\to \infty as n→∞n\to \infty. Can AA be partitioned into two disjoint additive bases of order 22?

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.

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