417.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 417

References:

erdosproblems.com/417
[Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.

Imports (1)

FormalConjecturesUtil

theorem Erdos417.erdos_417.parts.i Let$$V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$$and$$V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$$ Does $\lim V(x)/V'(x)$ exist? sorry line 41 source
Source documentation
Let$$V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$$and$$V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$$ Does $\lim V(x)/V'(x)$ exist?

depends on sorry what this means
Statement uses: Iff, True, Exists, ENNReal, Filter.Tendsto, Real, HDiv.hDiv, Set.ncard, Nat, Set.ofPred, And, Membership.mem, Set, Set.range, Nat.totient, LE.le, Set.image, OfNat.ofNat, Filter.atTop, nhds
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_417.parts.i :
    answer(sorry) ↔ ∃ L : ENNReal, Tendsto (fun x ↦
      ({ k | k ∈ range totient ∧ (k : ℝ) ≤ x }.ncard : ENNReal) /
      ((totient '' { m | 1 ≤ m ∧ (m : ℝ) ≤ x }).ncard : ENNReal))
      atTop (𝓝 L) := by
…
```
theorem Erdos417.erdos_417.parts.ii Is it $>1$? sorry line 52 source
Source documentation
Is it $>1$?

depends on sorry what this means
Statement uses: Iff, True, Exists, ENNReal, And, GT.gt, OfNat.ofNat, Filter.Tendsto, Real, HDiv.hDiv, Set.ncard, Nat, Set.ofPred, Membership.mem, Set, Set.range, Nat.totient, LE.le, Set.image, Filter.atTop and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_417.parts.ii :
    answer(sorry) ↔ ∃ L > (1 : ENNReal), Tendsto (fun x ↦
      ({ k | k ∈ range totient ∧ (k : ℝ) ≤ x }.ncard : ENNReal) /
      ((totient '' { m | 1 ≤ m ∧ (m : ℝ) ≤ x }).ncard : ENNReal))
      atTop (𝓝 L) := by
…
```