942.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 942

Reference: erdosproblems.com/942

Imports (1)

FormalConjecturesUtil

def Erdos942.erdos_942.h Let $h(n)$ count the number of powerful integers in $[n^2, (n + 1)^2)$ line 35 source
Source documentation
Let $h(n)$ count the number of powerful integers in $[n^2, (n + 1)^2)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, Nat.Powerful, Finset.Ico, HPow.hPow, OfNat.ofNat, HAdd.hAdd
10 instances, coercions and auxiliary constants left out.
```
def erdos_942.h (n : ℕ) : ℕ := ((Finset.Ico (n ^ 2) ((n + 1) ^ 2)).filter Powerful).card
…
```
theorem Erdos942.erdos_942 Is there some constant $c > 0$ such that $h(n) < (\log n)^{c + o(1)}$ and, for infinitely many $n$, $h(n) > (\log n)^{c - o(1)}$ sorry line 42 source
Source documentation
Is there some constant $c > 0$ such that $h(n) < (\log n)^{c + o(1)}$ and, for infinitely many $n$, $h(n) > (\log n)^{c - o(1)}$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Asymptotics.IsLittleO, Filter.atTop, Filter.Eventually, LT.lt, Erdos942.erdos_942.h, HPow.hPow, Real.log, HAdd.hAdd, Set.Infinite, Set.ofPred, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
33 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_942 : answer(sorry) ↔ ∃ c > 0, ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧
    (∀ᶠ n in atTop, erdos_942.h n < (Real.log n) ^ (c + o n)) ∧
    {n | erdos_942.h n > (Real.log n) ^ (c - o n)}.Infinite := by
…
```
Mentions: c, Erdos942.erdos_942.h
theorem Erdos942.erdos_942.variants.limsup It is not hard to prove that $\limsup h(n) = \infty$ sorry line 51 source
Source documentation
It is not hard to prove that $\limsup h(n) = \infty$.

depends on sorry what this means
Statement uses: Eq, ENat, Filter.limsup, Nat, Function.comp, Erdos942.erdos_942.h, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_942.variants.limsup :
    atTop.limsup (((fun (n : ℕ) ↦ (n : ℕ∞)) ∘ erdos_942.h)) = ⊤ := by
…
```
Mentions: Erdos942.erdos_942.h
theorem Erdos942.erdos_942.variants.density It is not hard to prove that the density $\delta_l$ of integers for which $h(n) = l$ exists and satisfies $$\sum_l \delta_l = 1$$ sorry line 60 source
Source documentation
It is not hard to prove that the density $\delta_l$ of integers for which $h(n) = l$ exists and satisfies $$\sum_l \delta_l = 1$$.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Set.HasDensity, Set.ofPred, Eq, Erdos942.erdos_942.h, Set.univ, tsum, SummationFilter.unconditional, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_942.variants.density :
    ∃ δ : ℕ → ℝ, ∀ l, {n | erdos_942.h n = l}.HasDensity (δ l) ∧
    ∑' l, δ l = 1 := by
…
```
Mentions: Erdos942.erdos_942.h