FormalConjectures.Wikipedia.Grimm

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import FormalConjecturesUtil

Grimm's conjecture

Reference: Wikipedia

open Functionnamespace Grimm

Grimm's Conjecture If $n, n+1, \dots, n+k-1$ are all composite numbers, then there are $k$ distinct primes $p_i$ such that $p_i$ divides $n + i$ for all $0 \le i \le k-1$.

@[category research open, AMS 11]
theorem grimm_conjecture (n k : ℕ) (hn : 1 ≤ n) (hk : 1 ≤ k)
    (h : ∀ i : Fin k, (n + i).Composite) :
    ∃ ps : Fin k ↪ ℕ,  ∀ i : Fin k, (ps i).Prime ∧ ps i ∣ (n + i) := byn:ℕk:ℕhn:1 ≤ nhk:1 ≤ kh:∀ (i : Fin k), (n + ↑i).Composite⊢ ∃ ps, ∀ (i : Fin k), Nat.Prime (ps i) ∧ ps i ∣ n + ↑i
  sorryAll goals completed! 🐙

Grimm's Conjecture, weaker version If $n, n+1, \dots, n+k-1$ are all composite numbers, then their product has at least $k$ distinct prime divisors.

@[category research open, AMS 11]
theorem grimm_conjecture_weak (n k : ℕ) (hn : 1 ≤ n) (hk : 1 ≤ k)
    (h : ∀ i : Fin k, (n + i).Composite) :
    ∃ ps : Fin k ↪ ℕ,  ∀ i : Fin k, (ps i).Prime ∧ ∃ j : Fin k, ps i ∣ (n + j) := byn:ℕk:ℕhn:1 ≤ nhk:1 ≤ kh:∀ (i : Fin k), (n + ↑i).Composite⊢ ∃ ps, ∀ (i : Fin k), Nat.Prime (ps i) ∧ ∃ j, ps i ∣ n + ↑j
  sorryAll goals completed! 🐙

end Grimm