Problem 330

Does there exist a minimal basis with positive density, say $A\subset\mathbb{N}$ , such that for any $n\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive?

Source: erdosproblems.com/330

Number theory
Additive bases

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN), the site's label. The standing rests on one accepted claim page, the AI-generated construction of 2026, accepted on the site's relabeling of 2026-05-11; the checks posted on the thread were ChatGPT runs, and there is no refereed publication. The label's Lean qualification dates from 2026-05-11, after the first outside formalization was announced (2026-05-05); a single-file vendoring of it followed; neither is built or audited in this corpus, and both are linked from the claim page.

Problem 330

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Proof claims

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