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module
public import Mathlib.Algebra.Polynomial.Degree.Defs
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Data.Nat.Prime.Defs@[expose] public sectionopen PolynomialThis file introduces some common assumptions made on polynomials in certain conjectures. Note that the terminology is non-standard.
Bunyakovsky condition A condition on polynomials in the Schinzel and Bunyakovsky conjectures. Holds for nonconstant irreducible polynomial with positive leading coefficient.
def BunyakovskyCondition (f : ℤ[X]) : Prop :=
1 ≤ f.degree ∧ Irreducible f ∧ 0 < f.leadingCoeffSchinzel condition A condition on sets of polynomials in the Schinzel and Bunyakovsky conjectures. Holds if for every prime $p$ there exists a natural $n$ such that for every polynomial $f$ in the set, $p$ does not divide $f(n)$.
def SchinzelCondition (fs : Finset ℤ[X]) : Prop :=
∀ p : ℕ, p.Prime → ∃ n : ℕ, ∀ f ∈ fs, ¬(p : ℤ) ∣ f.eval (n : ℤ)