Problem detail · source-aware
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Let $A\subset\mathbb{N}$ be infinite. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$? The question as posed follows negatively from Davenport–Erdős (1951). The AI result settles Tenenbaum's harder variant, also negatively: there is an infinite $A$ such that for every $k\geq 1$ the set of multiples of $A+k$ has upper density below $0.34$.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
A DeepMind prover agent constructed an infinite set $A$ such that for every $k\geq 1$ the set of multiples of $A+k$ has upper density less than $0.34$, resolving Tenenbaum's variant of the problem in the negative.
Provider: Google DeepMind · Prompt public: unknown · Independence: unknown
erdosproblems.com marks the problem DISPROVED and documents the DeepMind construction in the page remarks; the variant result is recorded there without a separate formal artifact.
Correctness: unknown · statement fidelity: unaudited · peer review: none
The question as posed was implicit in Davenport–Erdős (1951); the AI result settles Tenenbaum's open variant negatively
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.