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 statement4 of 9

Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ . If all elements in $A$ are pairwise coprime then $\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}/\log N$

FormalConjectures/ErdosProblems/12.leanErdos12.erdos_12.variants.baier3 linesExact file

1∀ (A : Set ℕ),2  Erdos12.IsGood A →3    A.Pairwise Nat.Coprime → (fun N => ↑(A ∩ Set.Icc 1 N).ncard) =O[Filter.atTop] fun N => ↑N ^ (2 / 3) / Real.log ↑N

SolvedStatement only, no proof

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