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import FormalConjecturesUtilErdős Problem 386
namespace Erdos386
open Nat
There is a $k$, such that $2 \le k \le n - 2$ and $\binom{n}{k}$ can be the product of consecutive primes infinitely often?
@[category research open, AMS 11]
theorem erdos_386 :
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 := ⊢ 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
All 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 := ⊢ 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
All 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 := ⊢ True ↔ ∃ᶠ (n : ℕ) in Filter.atTop, 2 ≤ n - 2 ∧ ∃ p q, n.choose 2 = ∏ i ∈ Finset.Ico p q, nth Nat.Prime i
All goals completed! 🐙
end Erdos386