208.lean`

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4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 208

Reference: erdosproblems.com/208

Imports (1)

FormalConjecturesUtil

def Erdos208.erdos208.s The sequence of squarefree numbers, denoted by s as in Erdős problem 208 line 32 source
Source documentation
The sequence of squarefree numbers, denoted by s as in Erdős problem 208.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, Squarefree
1 instances, coercions and auxiliary constants left out.
```
noncomputable def erdos208.s : ℕ → ℕ := Nat.nth Squarefree
…
```
theorem Erdos208.erdos_208.parts.i Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers sorry line 41 source
Source documentation
Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers. Is it true that for any $\epsilon > 0$ and large $n$, $s_{n+1} - s_n \ll_\epsilon s_n^\epsilon$?

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, HSub.hSub, Erdos208.erdos208.s, HAdd.hAdd, HPow.hPow
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_208.parts.i : answer(sorry) ↔
    ∀ ε > (0 : ℝ), (fun n => (s (n + 1) - s n : ℝ)) =O[atTop] (fun n => (s n : ℝ)^ε) := by sorry
…
```
theorem Erdos208.erdos_208.parts.ii Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers sorry line 49 source
Source documentation
Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers. Is it true that $s_{n + 1} - s_n \le (1 + o(1)) \cdot (\pi^2 / 6) \cdot \log (s_n) / \log (\log (s_n))$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, Filter.Eventually, LE.le, HSub.hSub, Erdos208.erdos208.s, HAdd.hAdd, HDiv.hDiv, HMul.hMul, HPow.hPow, Real.pi, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_208.parts.ii : answer(sorry) ↔ ∃ (c : ℕ → ℝ), (c =o[atTop] (1 : ℕ → ℝ)) ∧ ∀ᶠ n in atTop,
      s (n + 1) - s n ≤ (1 + (c n)) * (π^2 / 6) * log (s n) / log (log (s n)) := by
…
```
Mentions: c
theorem Erdos208.erdos_208.variants.log_bound In [Er79] Erdős says perhaps $s_{n+1} - s_n \ll \log s_n$, but he is 'very doubtful' sorry line 59 source
Source documentation
In [Er79] Erdős says perhaps $s_{n+1} - s_n \ll \log s_n$, but he is 'very doubtful'.
[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HSub.hSub, Erdos208.erdos208.s, HAdd.hAdd, OfNat.ofNat, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_208.variants.log_bound :
    (fun n ↦ (s (n + 1) - s n : ℝ)) =O[atTop] fun n ↦ log (s n) := by sorry
…
```