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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FormalConjectures/ErdosProblems/

978.lean

Retained formal statement4 of 6

If $k > 3$ (and $k \neq 2^l$ ), then are there infinitely many $n$ for which $f(n)$ is $(k-2)$ -power-free?

This was disproved by the DeepMind prover agent.

FormalConjectures/ErdosProblems/978.leanErdos978.erdos_978.variants.allow_fixed_divisors8 linesExact file

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

SolvedProof has a holeformal conjecturesexternal proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.

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