FormalConjectures/Paper/PrimeTuples.lean

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L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Prime Tuples Conjecture

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

Imports (1)
  1. theorem PrimeTuplesConjecture.prime_tuples_conjecture For any k ≥ 2, let a₁,...,aₖ and b₁,...,bₖ be integers with aᵢ > 0 sorry line 37 source

    Source documentation

    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.

    depends on sorry what this means

    Statement uses: Nat, LE.le, OfNat.ofNat, Fin, PNat, Nat.Prime, Exists, Not, Dvd.dvd, Finset.prod, Finset.univ, HAdd.hAdd, HMul.hMul, Set.Infinite, Set.ofPred

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    20 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem 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} := by sorry
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