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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FormalConjectures/ErdosProblems/

12.lean

Retained formal statement7 of 9

An example of an $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ and such that $\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N > 0$ is given by the set of $p^2$ , where $p\equiv 3\pmod{4}$ is prime.

FormalConjectures/ErdosProblems/12.leanErdos12.erdos_12.variants.example3 linesExact file

1∀ (A : Set ℕ),2  A = {x | ∃ p, ∃ (_ : Nat.Prime p) (_ : p ≡ 3 [MOD 4]), p ^ 2 = x} →3    Erdos12.IsGood A ∧ 0 < Filter.liminf (fun N => ↑(A ∩ Set.Icc 1 N).ncard * Real.log ↑N / √↑N) Filter.atTop

SolvedStatement only, no proof

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