FormalConjectures/ErdosProblems/1146.lean

GitHub source file

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1146

References:

Imports (1)
  1. def Erdos1146.IsEssentialComponent We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A \oplus B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density line 40 source

    Source documentation

    We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A \oplus B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density. Here, the sumset is the appropriate one for Schnirelmann density, $A \oplus B = \{a+b \mid a \in A \cup \{0\}, b \in B \cup \{0\}\}$ (i.e. $(A \cup \{0\}) + (B \cup \{0\})$). This avoids the trivial case where the sumset misses $1$ simply because neither $A$ nor $B$ contains $0$.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Set, Nat

    Proof uses: Real, schnirelmannDensity, Classical.propDecidable, Membership.mem, LT.lt, OfNat.ofNat, GT.gt, HAdd.hAdd, Union.union, Singleton.singleton

    12 instances, coercions and auxiliary constants left out.

    def IsEssentialComponent (A : Set ℕ) : Prop :=
…
  2. theorem Erdos1146.erdos_1146 Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component? sorry line 54 source

    Source documentation

    Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component?

    In [Ru99] Ruzsa states "The simplest set with a chance to be an essential component is the collection of numbers in the form $2^m3^n$ and Erdős often asked whether it is an essential component or not; I do not even have a plausible guess."

    depends on sorry what this means

    Statement uses: Iff, True, Erdos1146.IsEssentialComponent, Set.ofPred, Nat, Exists, Eq, HMul.hMul, HPow.hPow, OfNat.ofNat

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    14 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_1146 :
    answer(sorry) ↔ IsEssentialComponent { k | ∃ m n : ℕ, k = 2 ^ m * 3 ^ n } := by
…

    Mentions: Erdos1146.IsEssentialComponent