325.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 325

Reference: erdosproblems.com/325

Imports (1)

FormalConjecturesUtil

def Erdos325.IsSumThreePower A predicate for $n$ to be the sum of three $k$th powers line 32 source
Source documentation
A predicate for $n$ to be the sum of three $k$th powers.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, Eq, HAdd.hAdd, HPow.hPow
6 instances, coercions and auxiliary constants left out.
```
def IsSumThreePower (k n : ℕ) : Prop := ∃ a b c, a ^ k + b ^ k + c ^ k = n
```
Mentions: c
def Erdos325.cardIsSumThreePowerBelow The number of integers $\leq x$ which are the sum of three $k$th powers line 35 source
Source documentation
The number of integers $\leq x$ which are the sum of three $k$th powers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set.ncard, Set.ofPred, And, Membership.mem, Set, Set.Iic, Erdos325.IsSumThreePower
2 instances, coercions and auxiliary constants left out.
```
noncomputable def cardIsSumThreePowerBelow (k x : ℕ) : ℕ :=
…
```
theorem Erdos325.erdos_325 Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it true that $f_{k, 3}(x) \gg x ^ (3 / k)$? sorry line 43 source
Source documentation
Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it true that $f_{k, 3}(x) \gg x ^ (3 / k)$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Asymptotics.IsBigO, Real, Filter.atTop, HPow.hPow, HDiv.hDiv, Erdos325.cardIsSumThreePowerBelow
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_325 :
     answer(sorry) ↔ ∀ k : ℕ, 3 ≤ k → (fun x : ℕ => (x : ℝ) ^ (3 / k : ℝ)) =O[atTop]
      (fun x : ℕ => (cardIsSumThreePowerBelow k x : ℝ)) := by
…
```
Mentions: Erdos325.cardIsSumThreePowerBelow
theorem Erdos325.erdos_325.variants.weaker Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it even true that $f_{k, 3}(x) \gg_{\epsilon} x ^ (3 / k - \epsilon)$? sorry line 53 source
Source documentation
Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it even true that $f_{k, 3}(x) \gg_{\epsilon} x ^ (3 / k - \epsilon)$?

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Nat, LE.le, Asymptotics.IsBigO, Filter.atTop, HPow.hPow, HSub.hSub, HDiv.hDiv, Erdos325.cardIsSumThreePowerBelow
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 erdos_325.variants.weaker :
    answer(sorry) ↔ ∀ ε > 0, ∀ k : ℕ, 3 ≤ k → (fun x : ℕ => (x : ℝ) ^ ((3 / k : ℝ) - ε)) =O[atTop]
      (fun x => (cardIsSumThreePowerBelow k x : ℝ)) := by
…
```
Mentions: Erdos325.cardIsSumThreePowerBelow
theorem Erdos325.erdos_325.variants.wooley For $k = 3$, the best known is due to Wooley [Wo15], who proved $f_{3, 3}(x) \gg x^{0.91709477}$ sorry line 65 source
Source documentation
For $k = 3$, the best known is due to Wooley [Wo15], who proved $f_{3, 3}(x) \gg x^{0.91709477}$.
[Wo15] Wooley, Trevor D., Sums of three cubes, II. Acta Arith. (2015), 73-100.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HPow.hPow, OfScientific.ofScientific, Bool.true, Erdos325.cardIsSumThreePowerBelow, OfNat.ofNat
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_325.variants.wooley :
    (fun x : ℕ => (x : ℝ) ^ (0.91709477 : ℝ)) =O[atTop]
      (fun x => (cardIsSumThreePowerBelow 3 x : ℝ)) := by
…
```
Mentions: Erdos325.cardIsSumThreePowerBelow