FormalConjectures/ErdosProblems/1108.lean

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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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1108

Reference: erdosproblems.com/1108

Imports (1)
  1. def Erdos1108.FactorialSums The set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}_{\geq 1}\text{ finite}\right\}$ of all finite sums of distinct factorials line 37 source

    Source documentation

    The set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}_{\geq 1}\text{ finite}\right\}$ of all finite sums of distinct factorials. The indices are positive because $0! = 1!$, so allowing $0$ would count the value $1$ twice.

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

    Statement uses: Set, Nat

    Proof uses: Set.ofPred, Exists, Finset, And, Membership.mem, LT.lt, OfNat.ofNat, Eq, Finset.sum, Nat.factorial

    5 instances, coercions and auxiliary constants left out.

    def FactorialSums : Set ℕ :=
…
  2. theorem Erdos1108.erdos_1108.parts.i Documented declaration: i sorry line 44 source

    Source documentation

    For each $k \geq 2$, does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}_{\geq 1}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?

    depends on sorry what this means

    Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Set.Finite, Set.ofPred, And, Membership.mem, Set, Erdos1108.FactorialSums, Exists, Eq, HPow.hPow

    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_1108.parts.i : answer(sorry) ↔ ∀ k ≥ 2,
    Set.Finite { a | a ∈ FactorialSums ∧ ∃ m : ℕ, m ^ k = a } := by
…

    Mentions: Erdos1108.FactorialSums

  3. theorem Erdos1108.erdos_1108.parts.ii Does the set sorry line 53 source

    Source documentation

    Does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}_{\geq 1}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many powerful numbers (Nat.Powerful)?

    depends on sorry what this means

    Statement uses: Iff, True, Set.Finite, Nat, Set.ofPred, And, Membership.mem, Set, Erdos1108.FactorialSums, Nat.Powerful

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

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_1108.parts.ii :
     answer(sorry) ↔ {a ∈ FactorialSums | a.Powerful}.Finite := by
…

    Mentions: Erdos1108.FactorialSums