Erdős Problem 978

Reference:

• erdosproblems.com/978
• [Ho67] Hooley, C., On the power free values of polynomials. Mathematika (1967), 21--26.
• [Br11] Browning, T. D., Power-free values of polynomials. Arch. Math. (Basel) (2011), 139--150.
• [Er53] Erdős, P., Arithmetical properties of polynomials. J. London Math. Soc. (1953), 416--425.

theorem Erdos978.erdos_978.sub_one {f : Polynomial ℤ} (hi : Irreducible f) (hd : f.natDegree > 2) (hp : ¬∃ (l : ℕ), f.natDegree = 2 ^ l) : {n : ℕ | Powerfree (f.natDegree - 1) (Polynomial.eval (↑n) f)}.Infinite

Let f ∈ ℤ[X] be an irreducible polynomial. Suppose that the degree k of f is larger than 2 and is not equal to a power of 2. Then the set of n such that f n is (k - 1)-th power free is infinite, and this is proved in [Er53].

theorem Erdos978.erdos_978.sub_one_density {f : Polynomial ℤ} (hi : Irreducible f) (hd : f.natDegree > 2) (hp : ¬∃ (p : ℕ), Nat.Prime p ∧ ∀ (n : ℕ), ↑p ^ (f.natDegree - 1) ∣ Polynomial.eval (↑n) f) : {n : ℕ | Powerfree (f.natDegree - 1) (Polynomial.eval (↑n) f)}.HasPosDensity

Let f ∈ ℤ[X] be an irreducible polynomial. Suppose that the degree k of f is larger than 2, and f n have no fixed (k - 1)-th power divisors other than 1. Then the set of n such that f n is (k - 1)-th power free has positive density, and this is proved in [Ho67].

theorem Erdos978.erdos_978.sub_two {f : Polynomial ℤ} (hi : Irreducible f) (hd : f.natDegree ≥ 9) (hp : ¬∃ (p : ℕ), Nat.Prime p ∧ ∀ (n : ℕ), ↑p ^ (f.natDegree - 1) ∣ Polynomial.eval (↑n) f) : {n : ℕ | Powerfree (f.natDegree - 2) (Polynomial.eval (↑n) f)}.Infinite

If the degree k of f is larger than or equal to 9, then the set of n such that f n is (k - 2)-th power free has infinitely many elements. This result is proved in [Br11].

theorem Erdos978.erdos_978.sub_two' : sorry ↔ ∀ {f : Polynomial ℤ}, Irreducible f → f.natDegree > 2 → (¬∃ (p : ℕ), Nat.Prime p ∧ ∀ (n : ℕ), ↑p ^ (f.natDegree - 1) ∣ Polynomial.eval (↑n) f) → {n : ℕ | Powerfree (f.natDegree - 2) (Polynomial.eval (↑n) f)}.Infinite

Is it true that the set of n such that f n is (k - 2)-th power free has infinitely many elements?

theorem Erdos978.erdos_978.squarefree : sorry ↔ {n : ℕ | Squarefree (n ^ 4 + 2)}.Infinite

Does n ^ 4 + 2 represent infinitely many squarefree numbers?