46.lean`

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9 declarations, 5 contain sorry, 0 declared as axioms

Ben Green's Open Problem 46

What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \pmod{p}$, one for each prime $p \leq x$?

References:

[Gr24] Ben Green's Open Problem 46
[FGK18] Ford, K., Green, B., Konyagin, S., Maynard, J., & Tao, T. (2018). Long gaps between primes. Journal of the American Mathematical Society, 31(1), 65-105.
[Iw78] Iwaniec, Henryk. "On the problem of Jacobsthal." Demonstratio Mathematica 11.1 (1978): 225-232.

Imports (1)

FormalConjecturesUtil

def Green46.IsCoveredByResidues Given $x$ and $y$, can we cover the interval $[1, y]$ by residue classes $a_p \pmod p$ for each prime $p \le x$? line 41 source
Source documentation
Given $x$ and $y$, can we cover the interval $[1, y]$ by residue classes $a_p \pmod p$ for each prime $p \le x$?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, And, LE.le, Nat.Prime, Nat.ModEq
6 instances, coercions and auxiliary constants left out.
```
def IsCoveredByResidues (x y : ℕ) : Prop :=
…
```
def Green46.maxY The maximum $y$ for a given $x$, cast to a real number for asymptotics line 45 source
Source documentation
The maximum $y$ for a given $x$, cast to a real number for asymptotics.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: SupSet.sSup, Set.ofPred, Green46.IsCoveredByResidues
3 instances, coercions and auxiliary constants left out.
```
noncomputable def maxY (x : ℕ) : ℝ :=
…
```
def Green46.bestLower Best-known lower bound [Ra38] line 49 source
Source documentation
Best-known lower bound [Ra38].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HDiv.hDiv, HMul.hMul, Real.log
7 instances, coercions and auxiliary constants left out.
```
noncomputable def bestLower (x : ℕ) : ℝ :=
…
```
def Green46.bestUpper Best-known upper bound [Iw78] line 53 source
Source documentation
Best-known upper bound [Iw78].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HPow.hPow, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
noncomputable def bestUpper (x : ℕ) : ℝ := (x : ℝ) ^ 2
…
```
theorem Green46.green_46.improve_lower We conjecture that the best-known lower bound can be improved sorry line 57 source
Source documentation
We conjecture that the best-known lower bound can be improved.

depends on sorry what this means
Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsLittleO, Filter.atTop, Green46.bestLower, Asymptotics.IsBigO, Green46.maxY
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_46.improve_lower :
    let ans := (answer(sorry) : ℕ → ℝ)
…
```
theorem Green46.green_46.improve_upper We conjecture that the best-known upper bound can be improved sorry line 64 source
Source documentation
We conjecture that the best-known upper bound can be improved.

depends on sorry what this means
Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsLittleO, Filter.atTop, Green46.bestUpper, Asymptotics.IsBigO, Green46.maxY
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_46.improve_upper :
    let ans := (answer(sorry) : ℕ → ℝ)
…
```
theorem Green46.green_46.improve_upper_conjectured It seems very likely that we must have $y \ll x^{1+o(1)}$ [Gr24] sorry line 71 source
Source documentation
It seems very likely that we must have $y \ll x^{1+o(1)}$ [Gr24].

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, Asymptotics.IsBigO, Green46.maxY, HPow.hPow, HAdd.hAdd
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 green_46.improve_upper_conjectured :
    ∃ o : ℕ → ℝ, (o =o[atTop] fun _ : ℕ ↦ (1 : ℝ)) ∧
      maxY ≪ fun x ↦ (x : ℝ) ^ (1 + o x) := by
…
```
Mentions: Green46.maxY
theorem Green46.green_46.variants.lower Best known lower bound: $y \gg x \frac{\log x \log \log \log x}{\log \log x}$ [Ra38] sorry line 78 source
Source documentation
Best known lower bound: $y \gg x \frac{\log x \log \log \log x}{\log \log x}$ [Ra38].

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Green46.bestLower, Green46.maxY
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_46.variants.lower :
    bestLower ≪ maxY := by
…
```
Mentions: Green46.bestLower, Green46.maxY
theorem Green46.green_46.variants.upper Best known upper bound: $y \ll x^2$ [Iw78] sorry line 84 source
Source documentation
Best known upper bound: $y \ll x^2$ [Iw78].

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Green46.maxY, Green46.bestUpper
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_46.variants.upper :
    maxY ≪ bestUpper := by
…
```
Mentions: Green46.maxY, Green46.bestUpper