Erdős Problem #125

Full solution by DeepMind prover agent, formalized in Lean (30 Mar 2026). 2 AI contributions are recorded in all.

Problem
Erdős problem #125 (number theory, base representations); current status: disproved (Lean)
Credit
DeepMind prover agent · AI on its own
Checked
Lean formalizationListed in Tao's wiki
Timeline
First AI contribution 2026-02-21 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
SOLVEDPer Tao's AI-contributions wiki

The problem

Let A={∑ϵk3k:ϵk∈{0,1}}A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\} be the set of integers which have only the digits 0,10,1 when written base 33, and B={∑ϵk4k:ϵk∈{0,1}}B=\{ \sum\epsilon_k4^k : \epsilon_k\in \{0,1\}\} be the set of integers which have only the digits 0,10,1 when written base 44.

Does A+BA+B have positive lower density?

Statement quoted from erdosproblems.com/125, maintained by Thomas Bloom. Original source: [BEGL96][Er97,p.156]. 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
21 Feb, 2026 AI on its own DeepMind prover agent 🟡 Solution to variant problem (Lean)
30 Mar, 2026 AI on its own DeepMind prover agent 🟢 Full solution (Lean)

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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