Erdős problem 12 · problems.science

Let $A \subset \mathbb{N}$ be infinite with no distinct $a, b, c \in A$ such that $a \mid (b + c)$ with $b, c > a$ . Can $|A \cap [1, N]|/\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?

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10 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

12.lean

Retained formal statement1 of 9

Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ . Is there such an $A$ with $\liminf \frac{|A \cap \{1, \dotsc, N\}|}{N^{1/2}} > 0$ ?

The DeepMind prover agent has found a formal proof of this statement.

FormalConjectures/ErdosProblems/12.leanErdos12.erdos_12.parts.i1 lineExact file

1True ↔ ∃ A, Erdos12.IsGood A ∧ 0 < Filter.liminf (fun N => ↑(A ∩ Set.Icc 1 N).ncard / √↑N) Filter.atTop

SolvedProof has a holeformal conjecturesexternal 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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