323.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 323

Reference: erdosproblems.com/323

Imports (1)

FormalConjecturesUtil

def Erdos323.f Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers line 37 source
Source documentation
Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set.ncard, Set.ofPred, And, LE.le, Exists, Fin, Eq, Finset.sum, Finset.univ, HPow.hPow
7 instances, coercions and auxiliary constants left out.
```
noncomputable def f (k m x : ℕ) : ℕ :=
…
```
theorem Erdos323.erdos_323.parts.i Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all $\epsilon>0$? sorry line 47 source
Source documentation
Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all $\epsilon>0$?
This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Real, GT.gt, Asymptotics.IsBigO, Filter.atTop, HPow.hPow, HSub.hSub, Erdos323.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_323.parts.i :
    answer(sorry) ↔ ∀ k ≥ 1, ∀ ε > (0 : ℝ),
      (fun (x : ℕ) ↦ (x : ℝ) ^ (1 - ε)) =O[atTop] (fun (x : ℕ) ↦ (f k k x : ℝ)) := by
…
```
Mentions: Erdos323.f
theorem Erdos323.erdos_323.parts.ii Is it true that if $m < k$ then $f_{k,m}(x) \gg x^{m/k}$ for sufficiently large $x$? sorry line 56 source
Source documentation
Is it true that if $m < k$ then $f_{k,m}(x) \gg x^{m/k}$ for sufficiently large $x$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, LT.lt, Asymptotics.IsBigO, Real, Filter.atTop, HPow.hPow, HDiv.hDiv, Erdos323.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_323.parts.ii :
    answer(sorry) ↔ ∀ k m : ℕ, 1 ≤ m → m < k →
      (fun (x : ℕ) ↦ (x : ℝ) ^ ((m : ℝ) / (k : ℝ))) =O[atTop] (fun (x : ℕ) ↦ (f k m x : ℝ)) := by
…
```
Mentions: Erdos323.f
theorem Erdos323.erdos_323.variants.k_eq_2 The case $k=2$ was resolved by Landau, who showed $f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}$ for some constant $c>0$ sorry line 66 source
Source documentation
The case $k=2$ was resolved by Landau, who showed $f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}$ for some constant $c>0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsEquivalent, Nat, Filter.atTop, Erdos323.f, HDiv.hDiv, HMul.hMul, Real.sqrt, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_323.variants.k_eq_2 :
    ∃ c > 0, (fun (x : ℕ) ↦ (f 2 2 x : ℝ)) ~[atTop]
      (fun (x : ℕ) ↦ c * (x : ℝ) / Real.sqrt (Real.log (x : ℝ))) := by
…
```
Mentions: c, Erdos323.f
theorem Erdos323.erdos_323.variants.k_gt_2 For $k>2$ it is not known if $f_{k,k}(x)=o(x)$ sorry line 75 source
Source documentation
For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.

depends on sorry what this means
Statement uses: Iff, True, Nat, GT.gt, OfNat.ofNat, Asymptotics.IsLittleO, Real, Filter.atTop, Erdos323.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_323.variants.k_gt_2 :
    answer(sorry) ↔ ∀ k > 2, (fun (x : ℕ) ↦ (f k k x : ℝ)) =o[atTop] (fun (x : ℕ) ↦ (x : ℝ)) := by
…
```
Mentions: Erdos323.f