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Erdős problem 978

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Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient. Suppose that the degree k of f is larger than 2, is not equal to a power of 2, and f n has 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].

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978.lean

Retained formal statement6 of 6

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].

FormalConjectures/ErdosProblems/978.leanErdos978.erdos_978.variants.sub_two5 linesExact file
∀ {f : Polynomial ℤ},  Irreducible f →    9 ≤ f.natDegree →      (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬↑p ^ (f.natDegree - 1) ∣ Polynomial.eval (↑n) f) →        {n | Powerfree (f.natDegree - 2) (Polynomial.eval (↑n) f)}.Infinite
SolvedStatement only, no proof