FormalConjectures/ErdosProblems/386.lean

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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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 386

Reference: erdosproblems.com/386

Imports (1)
  1. theorem Erdos386.erdos_386 Let $2 \le k \le n - 2$ sorry line 38 source

    Source documentation

    Let $2 \le k \le n - 2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? Here $k$ may vary with $n$: the question asks for infinitely many admissible binomial coefficients, not for a single $k$ that works infinitely often.

    depends on sorry what this means

    Statement uses: Iff, True, Filter.Frequently, Nat, Exists, And, GE.ge, OfNat.ofNat, LE.le, HSub.hSub, Eq, Nat.choose, Finset.prod, Finset.Ico, Nat.nth, Nat.Prime, Filter.atTop

    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_386 :
    answer(sorry) ↔ ∃ᶠ n in .atTop, ∃ k ≥ 2,
      k ≤ n - 2 ∧ ∃ p q : ℕ, n.choose k = ∏ i ∈ .Ico p q, nth Nat.Prime i := by
…
  2. theorem Erdos386.erdos_386.variants.forall For all $2 \le k \le n - 2$, can $\binom{n}{k}$ be the product of consecutive primes infinitely often? sorry line 48 source

    Source documentation

    For all $2 \le k \le n - 2$, can $\binom{n}{k}$ be the product of consecutive primes infinitely often?

    depends on sorry what this means

    Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Filter.Frequently, And, LE.le, HSub.hSub, Exists, Eq, Nat.choose, Finset.prod, Finset.Ico, Nat.nth, Nat.Prime, Filter.atTop

    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_386.variants.forall :
    answer(sorry) ↔ ∀ k ≥ 2, ∃ᶠ n in .atTop,
      k ≤ n - 2 ∧ ∃ p q : ℕ, n.choose k = ∏ i ∈ .Ico p q, nth Nat.Prime i := by
…
  3. theorem Erdos386.erdos_386.variants.two Can $\binom{n}{2}$ be the product of consecutive primes infinitely often? sorry line 57 source

    Source documentation

    Can $\binom{n}{2}$ be the product of consecutive primes infinitely often?

    depends on sorry what this means

    Statement uses: Iff, True, Filter.Frequently, Nat, And, LE.le, OfNat.ofNat, HSub.hSub, Exists, Eq, Nat.choose, Finset.prod, Finset.Ico, Nat.nth, Nat.Prime, Filter.atTop

    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_386.variants.two :
    answer(sorry) ↔ ∃ᶠ n in .atTop,
      2 ≤ n - 2 ∧ ∃ p q : ℕ, n.choose 2 = ∏ i ∈ .Ico p q, nth Nat.Prime i := by
…