FormalConjectures.ErdosProblems.«386»

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module

public import FormalConjecturesUtil

Erdős Problem 386

Reference: erdosproblems.com/386

@[expose] public sectionnamespace Erdos386open Nat

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.

@[category research open, AMS 11]
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⊢ True ↔ ∃ᶠ (n : ℕ) in Filter.atTop, ∃ k ≥ 2, k ≤ n - 2 ∧ ∃ p q, n.choose k = ∏ i ∈ Finset.Ico p q, nth Nat.Prime i
    sorryAll goals completed! 🐙

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

@[category research open, AMS 11]
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⊢ True ↔ ∀ k ≥ 2, ∃ᶠ (n : ℕ) in Filter.atTop, k ≤ n - 2 ∧ ∃ p q, n.choose k = ∏ i ∈ Finset.Ico p q, nth Nat.Prime i
    sorryAll goals completed! 🐙

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

@[category research open, AMS 11]
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⊢ True ↔ ∃ᶠ (n : ℕ) in Filter.atTop, 2 ≤ n - 2 ∧ ∃ p q, n.choose 2 = ∏ i ∈ Finset.Ico p q, nth Nat.Prime i
    sorryAll goals completed! 🐙

end Erdos386