887.lean`

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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 887

References:

erdosproblems.com/887
[ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Erdos887.erdos_887.parts.i Is there an absolute constant sorry line 38 source
Source documentation
Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}})$.

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Finset.card, Finset.filter, Dvd.dvd, Finset.Ioo, Nat.floor, Real.sqrt, Nat.ceil, HAdd.hAdd, HMul.hMul, HPow.hPow, HDiv.hDiv, sorryAx, Lean.Name, Bool.false and 4 more
62 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_887.parts.i : ∀ C > (0 : ℝ), ∀ᶠ n in atTop,
    #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n } ≤ answer(sorry) := by
…
```
theorem Erdos887.erdos_887.parts.ii Is there an absolute constant sorry line 47 source
Source documentation
Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}})$.

depends on sorry what this means
Statement uses: Exists, Nat, Real, GT.gt, OfNat.ofNat, Filter.Eventually, LE.le, Finset.card, Finset.filter, Dvd.dvd, Finset.Ioo, Nat.floor, Real.sqrt, Nat.ceil, HAdd.hAdd, HMul.hMul, HPow.hPow, HDiv.hDiv, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
62 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_887.parts.ii : ∃ K, ∀ C > (0 : ℝ), ∀ᶠ n in atTop,
    #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n } ≤ K := by
…
```
theorem Erdos887.erdos_887.variants.rosenfeld_infinite A question of Erdős and Rosenfeld, who proved that there are infinitely many $n$ with (at least) $4$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + cn^{\frac{1}{4}})$ sorry line 56 source
Source documentation
A question of Erdős and Rosenfeld, who proved that there are infinitely many $n$ with (at least) $4$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + cn^{\frac{1}{4}})$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Infinite, Nat, Set.ofPred, LE.le, Finset.card, Finset.filter, Dvd.dvd, Finset.Ioo, Nat.floor, Real.sqrt, Nat.ceil, HAdd.hAdd, HMul.hMul, HPow.hPow, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
64 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_887.variants.rosenfeld_infinite : ∃ C > (0 : ℝ),
    Infinite {n : ℕ | 4 ≤ #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n }} := by
…
```
theorem Erdos887.erdos_887.variants.rosenfeld_4 Erdős and Rosenfeld, ask whether $4$ is the best possible $K$ for the infinitude of $n$ with (at least) $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + n^{\frac{1}{4}})$ sorry line 65 source
Source documentation
Erdős and Rosenfeld, ask whether $4$ is the best possible $K$ for the infinitude of $n$ with (at least) $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + n^{\frac{1}{4}})$.

depends on sorry what this means
Statement uses: IsGreatest, Nat, Set.ofPred, Exists, Real, And, GT.gt, OfNat.ofNat, Infinite, LE.le, Finset.card, Finset.filter, Dvd.dvd, Finset.Ioo, Nat.floor, Real.sqrt, Nat.ceil, HAdd.hAdd, HMul.hMul, HPow.hPow and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
64 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_887.variants.rosenfeld_4 :
    IsGreatest {K | ∃ C > (0 : ℝ),
      Infinite {n : ℕ | K ≤ #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n }}} 4 := by
…
```