945.lean`

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

9 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 945

References:

erdosproblems.com/945
[ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271.

Imports (1)

FormalConjecturesUtil

abbrev Erdos945.τ line 34 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Nat.divisors
def Erdos945.F Let $F(x)$ be the maximal $k$ such that there exist $n+1, \dots, n+k \le x$ with $τ(n+1), \dots, τ(n+k)$ all distinct, where $τ(m)$ counts the divisors of $m$ line 39 source
Source documentation
Let $F(x)$ be the maximal $k$ such that there exist $n+1, \dots, n+k \le x$ with $τ(n+1), \dots, τ(n+k)$ all distinct, where $τ(m)$ counts the divisors of $m$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, And, LE.le, HAdd.hAdd, Set.InjOn, Erdos945.τ, Set.Ioc
8 instances, coercions and auxiliary constants left out.
```
noncomputable def F (x : ℝ) : ℕ :=
…
```
def Erdos945.Erdos945Prop Implementation note: we define a Prop here and below to be able to easily formulate the equivalence between the two variants line 45 source
Source comment
Implementation note: we define a Prop here and below to be able to easily formulate the equivalence between the two variants. Because the theorems require `answer(sorry)` we can't handle this with `type_of%`.

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Exists, Real, And, Asymptotics.IsBigO, Filter.atTop, OfNat.ofNat, Filter.Eventually, LE.le, Erdos945.F, HPow.hPow, Real.log
10 instances, coercions and auxiliary constants left out.
```
def Erdos945Prop : Prop := ∃ O : ℝ → ℝ, O =O[atTop] (1 : ℝ → ℝ) ∧ ∀ᶠ x in atTop, F x ≤ log x ^ O x
…
```
Mentions: Erdos945.F
theorem Erdos945.erdos_945 Is it true that $F(x) \leq (\log x)^{O(1)}$? sorry line 51 source
Source documentation
Is it true that $F(x) \leq (\log x)^{O(1)}$?

depends on sorry what this means
Statement uses: Iff, True, Erdos945.Erdos945Prop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_945 : answer(sorry) ↔ Erdos945Prop := by
…
```
Mentions: Erdos945.Erdos945Prop
def Erdos945.Erdos945Constant line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Ne, Membership.mem, Set, Set.Icc, HAdd.hAdd, HPow.hPow, Real.log, Eq, Erdos945.τ, Filter.atTop
11 instances, coercions and auxiliary constants left out.
theorem Erdos945.erdos_945.variants.constant Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\log x)C]$ contains two integers with the same number of divisors? sorry line 66 source
Source documentation
Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\log x)C]$ contains two integers with the same number of divisors?

depends on sorry what this means
Statement uses: Iff, True, Erdos945.Erdos945Constant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_945.variants.constant : answer(sorry) ↔ Erdos945Constant := by
…
```
Mentions: Erdos945.Erdos945Constant
theorem Erdos945.erdos_945.variants.equivalence The two ways of phrasing the conjecture are equivalent sorry line 74 source
Source documentation
The two ways of phrasing the conjecture are equivalent.

depends on sorry what this means
Statement uses: Iff, Erdos945.Erdos945Prop, Erdos945.Erdos945Constant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_945.variants.equivalence : Erdos945Prop ↔ Erdos945Constant := by
…
```
Mentions: Erdos945.Erdos945Prop, Erdos945.Erdos945Constant
theorem Erdos945.erdos_945.variants.lower_bound Erdős and Mirsky [ErMi52] proved that $\frac{(\log x)^{1/2}}{\log\log x}\ll F(x)$ sorry line 81 source
Source documentation
Erdős and Mirsky [ErMi52] proved that $\frac{(\log x)^{1/2}}{\log\log x}\ll F(x)$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HDiv.hDiv, Real.sqrt, Real.log, Erdos945.F
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_945.variants.lower_bound :
    (fun (x : ℕ) => (log x).sqrt /(log x).log) =O[atTop] fun (n : ℕ) => (F n : ℝ) := by
…
```
Mentions: Erdos945.F
theorem Erdos945.erdos_945.variants.upper_bound Erdős and Mirsky [ErMi52] proved that $\log F(x) \ll \frac{(\log x)^{1/2}}$ sorry line 89 source
Source documentation
Erdős and Mirsky [ErMi52] proved that $\log F(x) \ll \frac{(\log x)^{1/2}}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Real.log, Erdos945.F, HDiv.hDiv, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_945.variants.upper_bound :
    (fun (n : ℕ) => (F n : ℝ).log) =O[atTop] fun (x : ℕ) => (log x).sqrt /(log x).log := by
…
```
Mentions: Erdos945.F