Erdős problem 978 · problems.science

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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6 retained statements2415f78e850a

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978.lean

Retained formal statement2 of 6

If $k>3$ (and $k \neq 2^l$ ), and for all primes $p$ there exists $n$ such that $p^{k-2}\nmid f(n)$ , then are there infinitely many $n$ for which $f(n)$ is $(k-2)$ -power-free?

FormalConjectures/ErdosProblems/978.leanErdos978.erdos_978.parts.ii8 linesExact file

1True ↔2  ∀ {f : Polynomial ℤ},3    Irreducible f →4      f.natDegree > 3 →5        (¬∃ l, f.natDegree = 2 ^ l) →6          0 < f.leadingCoeff →7            (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬↑p ^ (f.natDegree - 2) ∣ Polynomial.eval (↑n) f) →8              {n | Powerfree (f.natDegree - 2) (Polynomial.eval (↑n) f)}.Infinite

OpenStatement only, no proof

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