FormalConjectures.Paper.PrimeTuples

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import FormalConjecturesUtil

Prime Tuples Conjecture

Reference: [FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers (2007).


open Nat


namespace PrimeTuplesConjecture

For any k ≥ 2, let a₁,...,aₖ and b₁,...,bₖ be integers with aᵢ > 0. Suppose that for every prime p there exists an integer n such that p ∤ ∏ i, (aᵢ n + bᵢ). Then there exist infinitely many n such that aᵢ n + bᵢ is prime for all i.

@[category research open, AMS 11]
theorem declaration uses 'sorry'prime_tuples_conjecture {k : ℕ} (hk : 2 ≤ k) (a : Fin k → ℕ+) (b : Fin k → ℕ)
    (hab : ∀ p, p.Prime → ∃ n, ¬ p ∣ ∏ i, (a i * n + b i)) :
    Set.Infinite {n | ∀ i : Fin k, (a i * n + b i).Prime} := byk:ℕhk:2 ≤ ka:Fin k → ℕ+b:Fin k → ℕhab:∀ (p : ℕ), Nat.Prime p → ∃ n, ¬p ∣ ∏ i, (↑(a i) * n + b i)⊢ {n | ∀ (i : Fin k), Nat.Prime (↑(a i) * n + b i)}.Infinite sorryAll goals completed! 🐙


end PrimeTuplesConjecture