66.lean`

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L4DEL: Lean4 Domain Explanation Language

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

3 declarations, 2 contain sorry, 0 declared as axioms

Ben Green's Open Problem 66

Reference: Ben Green's Open Problem 66

Imports (1)

FormalConjecturesUtil

def Green66.IsSumOfTwoSquares A natural number is a sum of two squares if it can be written as a ^ 2 + b ^ 2 line 33 source
Source documentation
A natural number is a sum of two squares if it can be written as a ^ 2 + b ^ 2.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def IsSumOfTwoSquares (n : ℕ) : Prop :=
…
```
theorem Green66.green_66 Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$? sorry line 41 source
Source documentation
Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$? We formalize this as an eventual statement for sufficiently large real $X$.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Real, Exists, Nat, And, Green66.IsSumOfTwoSquares, Membership.mem, Set, Set.Icc, HSub.hSub, HMul.hMul, HDiv.hDiv, OfNat.ofNat, HPow.hPow, Filter.atTop
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 green_66 :
    answer(sorry) ↔
      ∀ᶠ X : ℝ in atTop,
        ∃ n : ℕ, IsSumOfTwoSquares n ∧
          (n : ℝ) ∈ Set.Icc (X - (1 / 10 : ℝ) * X ^ (1 / 4 : ℝ)) X := by
…
```
Mentions: Green66.IsSumOfTwoSquares
theorem Green66.green_66.variants.trivial_bound Green notes that a well-known, almost trivial argument gives an $O(X^{1/4})$ bound on the left sorry line 52 source
Source documentation
Green notes that a well-known, almost trivial argument gives an $O(X^{1/4})$ bound on the left.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Green66.IsSumOfTwoSquares, Membership.mem, Set, Set.Icc, HSub.hSub, HMul.hMul, HPow.hPow, HDiv.hDiv, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_66.variants.trivial_bound :
    ∃ C > (0 : ℝ), ∀ᶠ X : ℝ in atTop,
      ∃ n : ℕ, IsSumOfTwoSquares n ∧
        (n : ℝ) ∈ Set.Icc (X - C * X ^ (1 / 4 : ℝ)) X := by
…
```
Mentions: Green66.IsSumOfTwoSquares