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 statement2 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 $b,c > a$ . Does there exist some absolute constant $c > 0$ such that there are always infinitely many $N$ with $|A \cap \{1, \dotsc, N\}| < N^{1−c}$ ?

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

FormalConjectures/ErdosProblems/12.leanErdos12.erdos_12.parts.ii1 lineExact file

1False ↔ ∃ c > 0, ∀ (A : Set ℕ), Erdos12.IsGood A → {N | ↑(A ∩ Set.Icc 1 N).ncard < ↑N ^ (1 - c)}.Infinite

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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