FormalConjectures/GreensOpenProblems/27.lean

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L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

11 declarations, 7 contain sorry, 0 declared as axioms

Green's Open Problem 27

References:

Imports (1)
  1. def Green27.m Documented declaration: m line 39 source

    Source documentation

    This is $m(p)$ in [Be23]: the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat, Real

    Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, ZMod, LE.le, OfNat.ofNat, Finset.card, HasNoUniqueRepresentation, Eq

    9 instances, coercions and auxiliary constants left out.

    noncomputable def m (p : ℕ) : ℝ :=
…
  2. def Green27.primesAtTop atTop restricted to prime numbers line 43 source

    Source documentation

    atTop restricted to prime numbers.

    standard axioms: Quot.sound, propext what this means

    Statement uses: Filter, Nat

    Proof uses: Min.min, Filter.atTop, Filter.principal, Set.ofPred, Nat.Prime

    2 instances, coercions and auxiliary constants left out.

    def primesAtTop : Filter ℕ := atTop ⊓ 𝓟 {p : ℕ | p.Prime}
  3. def Green27.lowerBest Best-known lower bound [Be23, Theorem 3] line 46 source

    Source documentation

    Best-known lower bound [Be23, Theorem 3].

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat, Real

    Proof uses: HMul.hMul, HDiv.hDiv, Real.sqrt, Real.log

    7 instances, coercions and auxiliary constants left out.

    noncomputable def lowerBest (p : ℕ) : ℝ :=
…
  4. def Green27.upperBest Best-known upper bound [Be23, Theorem 5] line 51 source

    Source documentation

    Best-known upper bound [Be23, Theorem 5].

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat, Real

    Proof uses: HPow.hPow, Real.log, OfNat.ofNat

    7 instances, coercions and auxiliary constants left out.

    noncomputable def upperBest (p : ℕ) : ℝ := (Real.log (p : ℝ)) ^ 2
…
  5. theorem Green27.green_27.equivalent What is the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation? sorry line 58 source

    Source documentation

    What is the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?

    depends on sorry what this means

    Statement uses: Asymptotics.IsEquivalent, Nat, Real, Green27.primesAtTop, sorryAx, Bool.false, Green27.m

    Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    11 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_27.equivalent :
  (answer(sorry) : ℕ → ℝ) ~[primesAtTop] m := by
…

    Mentions: Green27.primesAtTop, Green27.m

  6. theorem Green27.green_27.lower Propose a better lower bound along primes sorry line 64 source

    Source documentation

    Propose a better lower bound along primes.

    depends on sorry what this means

    Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsLittleO, Green27.primesAtTop, Green27.lowerBest, Asymptotics.IsBigO, Green27.m

    Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_27.lower :
    let ans := (answer(sorry) : ℕ → ℝ)
…
  7. theorem Green27.green_27.upper Propose a better upper bound along primes sorry line 71 source

    Source documentation

    Propose a better upper bound along primes.

    depends on sorry what this means

    Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsLittleO, Green27.primesAtTop, Green27.upperBest, Asymptotics.IsBigO, Green27.m

    Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_27.upper :
    let ans := (answer(sorry) : ℕ → ℝ)
…
  8. theorem Green27.green_27.variants.lower_be23 We have $m(p) \geq \omega(p) \log p$ for some function $\omega(p)$ tending to infinity [Be23, Theorem 3] sorry line 80 source

    Source documentation

    We have $m(p) \geq \omega(p) \log p$ for some function $\omega(p)$ tending to infinity [Be23, Theorem 3].

    depends on sorry what this means

    Statement uses: Exists, Nat, Real, And, Filter.Tendsto, Green27.primesAtTop, Filter.atTop, Filter.Eventually, LE.le, HMul.hMul, Real.log, Green27.m

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    13 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_27.variants.lower_be23 :
  ∃ ω : ℕ → ℝ, Tendsto ω primesAtTop atTop ∧
    ∀ᶠ p in primesAtTop,
      ω p * Real.log (p : ℝ) ≤ m p := by
…

    Mentions: Green27.primesAtTop, Green27.m

  9. theorem Green27.green_27.variants.upper_be23 sorry line 89 source

    depends on sorry what this means

    Statement uses: Asymptotics.IsBigO, Nat, Real, Green27.primesAtTop, Green27.m, Green27.upperBest

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

  10. theorem Green27.green_27.variants.previous_lower Previous best-known lower bound $\log p \ll m(p)$ from [St76] sorry line 95 source

    Source documentation

    Previous best-known lower bound $\log p \ll m(p)$ from [St76].

    depends on sorry what this means

    Statement uses: Asymptotics.IsBigO, Nat, Real, Green27.primesAtTop, Real.log, Green27.m

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    7 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_27.variants.previous_lower :
  (fun p ↦ Real.log (p : ℝ)) =O[primesAtTop] m := by
…

    Mentions: Green27.primesAtTop, Green27.m

  11. theorem Green27.green_27.variants.previous_upper Previous best-known upper bound $m(p) \ll \sqrt{p}$ from [Be23] sorry line 101 source

    Source documentation

    Previous best-known upper bound $m(p) \ll \sqrt{p}$ from [Be23].

    depends on sorry what this means

    Statement uses: Asymptotics.IsBigO, Nat, Real, Green27.primesAtTop, Green27.m, Real.sqrt

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    7 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_27.variants.previous_upper :
  m =O[primesAtTop] (fun p ↦ Real.sqrt (p : ℝ)) := by
…

    Mentions: Green27.m, Green27.primesAtTop