Prime Tuples Conjecture #
Reference: [FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers (2007).
theorem
PrimeTuplesConjecture.prime_tuples_conjecture
{k : ℕ}
(hk : 2 ≤ k)
(a : Fin k → ℕ+)
(b : Fin k → ℕ)
(hab : ∀ (p : ℕ), Nat.Prime p → ∃ (n : ℕ), ¬p ∣ ∏ i : Fin k, (↑(a i) * n + b i))
:
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.