Erdős problem 26 · problems.science

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 $0.34$ .

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11 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

26.lean

Retained formal statement1 of 10

Let $A\subset\mathbb{N}$ be infinite such that $\sum_{a \in A} \frac{1}{a} = \infty$ . Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$ ?

This was formalized in Lean by Alexeev using Aristotle.

FormalConjectures/ErdosProblems/26.leanErdos26.erdos_261 lineExact file

1False ↔ ∀ (A : ℕ → ℕ), StrictMono A → Erdos26.IsThick A → ∃ k, Erdos26.IsBehrend fun x => A x + k

SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.

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