978.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

6 declarations, 6 contain sorry, 0 declared as axioms

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.

Imports (1)

FormalConjecturesUtil

theorem Erdos978.erdos_978.variants.sub_one Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient sorry line 40 source
Source documentation
Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient. 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].

depends on sorry what this means
Statement uses: Polynomial, Int, Irreducible, LT.lt, Nat, OfNat.ofNat, Polynomial.natDegree, Ne, HPow.hPow, Polynomial.leadingCoeff, Set.Infinite, Set.ofPred, Powerfree, HSub.hSub, Polynomial.eval
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_978.variants.sub_one {f : ℤ[X]} (hi : Irreducible f) (hd : 2 < f.natDegree)
    (hp : ∀ (x : ℕ), f.natDegree ≠ 2 ^ x) (hlc : 0 < f.leadingCoeff) :
    {n : ℕ | Powerfree (f.natDegree - 1) (f.eval (n : ℤ))}.Infinite := by
…
```
Mentions: Powerfree
theorem Erdos978.erdos_978.parts.i Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient sorry line 50 source
Source documentation
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].

depends on sorry what this means
Statement uses: Polynomial, Int, Irreducible, LT.lt, Nat, OfNat.ofNat, Polynomial.natDegree, Ne, HPow.hPow, Polynomial.leadingCoeff, Nat.Prime, Exists, Not, Dvd.dvd, HSub.hSub, Polynomial.eval, Set.HasPosDensity, Set.ofPred, Powerfree, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_978.parts.i {f : ℤ[X]} (hi : Irreducible f) (hd : 2 < f.natDegree)
    (hp2 : ∀ (x : ℕ), f.natDegree ≠ 2 ^ x) (hlc : 0 < f.leadingCoeff)
    (hp : ∀ (p : ℕ), p.Prime → ∃ n : ℕ, ¬ (p : ℤ) ^ (f.natDegree - 1) ∣ f.eval (n : ℤ)) :
    HasPosDensity {n : ℕ | Powerfree (f.natDegree - 1) (f.eval (n : ℤ))} := by
…
```
Mentions: Powerfree
theorem Erdos978.erdos_978.variants.sub_two Documented declaration: sub_two sorry line 60 source
Source documentation
If the degree k of f is larger than or equal to 9, and f n has no fixed (k - 2)-th power divisors other than 1, 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].

depends on sorry what this means
Statement uses: Polynomial, Int, Irreducible, LE.le, Nat, OfNat.ofNat, Polynomial.natDegree, Nat.Prime, Exists, Not, Dvd.dvd, HPow.hPow, HSub.hSub, Polynomial.eval, Set.Infinite, Set.ofPred, Powerfree
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_978.variants.sub_two {f : ℤ[X]} (hi : Irreducible f) (hd : 9 ≤ f.natDegree)
    (hp : ∀ (p : ℕ), p.Prime → ∃ n : ℕ, ¬ (p : ℤ) ^ (f.natDegree - 2) ∣ f.eval (n : ℤ)) :
    {n : ℕ | Powerfree (f.natDegree - 2) (f.eval (n : ℤ))}.Infinite := by
…
```
Mentions: Powerfree
theorem Erdos978.erdos_978.variants.allow_fixed_divisors sorry line 73 source
depends on sorry what this means
Statement uses: Iff, False, Polynomial, Int, Irreducible, GT.gt, Nat, Polynomial.natDegree, OfNat.ofNat, Not, Exists, Eq, HPow.hPow, LT.lt, Polynomial.leadingCoeff, And, Nat.Prime, Dvd.dvd, HSub.hSub, Polynomial.eval and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos978.erdos_978.parts.ii 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? sorry line 85 source
Source documentation
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?

depends on sorry what this means
Statement uses: Iff, True, Polynomial, Int, Irreducible, GT.gt, Nat, Polynomial.natDegree, OfNat.ofNat, Not, Exists, Eq, HPow.hPow, LT.lt, Polynomial.leadingCoeff, Nat.Prime, Dvd.dvd, HSub.hSub, Polynomial.eval, Set.Infinite and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_978.parts.ii : answer(sorry) ↔
    ∀ {f : ℤ[X]}, Irreducible f → f.natDegree > 3 →
    (¬ ∃ l : ℕ, f.natDegree = 2 ^ l) → 0 < f.leadingCoeff →
    (∀ (p : ℕ), p.Prime → ∃ n : ℕ, ¬ (p : ℤ) ^ (f.natDegree - 2) ∣ f.eval (n : ℤ)) →
    {n : ℕ | Powerfree (f.natDegree - 2) (f.eval (n : ℤ))}.Infinite := by
…
```
Mentions: Powerfree
theorem Erdos978.erdos_978.parts.iii Does n ^ 4 + 2 represent infinitely many squarefree numbers? sorry line 94 source
Source documentation
Does n ^ 4 + 2 represent infinitely many squarefree numbers?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Squarefree, HAdd.hAdd, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_978.parts.iii : answer(sorry) ↔ {n : ℕ | Squarefree (n ^ 4 + 2)}.Infinite := by
…
```