829.lean`

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

10 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 829

References:

erdosproblems.com/829
[Er83] Erdős, P. and Dudley, U., Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298.

Imports (1)

FormalConjecturesUtil

def Erdos829.cubes The set of perfect cubes in $\mathbb{N}$ line 36 source
Source documentation
The set of perfect cubes in $\mathbb{N}$.

standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Eq, HPow.hPow, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def cubes : Set ℕ := {n | ∃ k, k ^ 3 = n}
…
```
lemma Erdos829.mem_cubes_iff Membership in cubes can be witnessed by a bounded cube root, which makes it decidable for concrete values line 41 source
Source documentation
Membership in cubes can be witnessed by a bounded cube root, which makes it decidable for concrete values.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Iff, Membership.mem, Set, Erdos829.cubes, Exists, And, LT.lt, HAdd.hAdd, OfNat.ofNat, Eq, HPow.hPow
Proof uses: Iff.intro, Exists.intro, And.intro, LE.le, Nat.le_self_pow, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Decidable.byContradiction, Not, rfl
26 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
lemma mem_cubes_iff (m : ℕ) : m ∈ cubes ↔ ∃ k, k < m + 1 ∧ k ^ 3 = m := by
…
```
Mentions: Erdos829.cubes
theorem Erdos829.erdos_829 Erdős Problem 829 (open) sorry line 57 source
Source documentation
Erdős Problem 829 (open). Let $A \subseteq \mathbb{N}$ be the set of perfect cubes. Is it true that $(1_A \ast 1_A)(n) \ll (\log n)^{O(1)}$? That is, does there exist a natural number $C$ such that the number of representations of $n$ as a sum of two cubes is $O((\log n)^C)$ as $n \to \infty$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Asymptotics.IsBigO, Real, Filter.atTop, AdditiveCombinatorics.sumRep, Erdos829.cubes, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_829 :
    answer(sorry) ↔
      ∃ C : ℕ, (fun n : ℕ => (sumRep cubes n : ℝ)) =O[atTop]
        (fun n : ℕ => (Real.log n) ^ C) := by
…
```
Mentions: Erdos829.cubes
theorem Erdos829.sumRep_cubes_zero There is exactly one ordered pair of cubes summing to $0$, namely $(0, 0)$ line 65 source
Source documentation
There is exactly one ordered pair of cubes summing to $0$, namely $(0, 0)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, AdditiveCombinatorics.sumRep, Erdos829.cubes, OfNat.ofNat
Proof uses: Eq.mpr, Finset.card, Prod, Finset.filter, And, Exists, LT.lt, HAdd.hAdd, Prod.fst, HPow.hPow, Prod.snd, Finset.HasAntidiagonal.antidiagonal, id, congrFun', congrArg, Eq.trans, Membership.mem, Set, Classical.propDecidable, AdditiveCombinatorics.sumRep_def and 9 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem sumRep_cubes_zero : sumRep cubes 0 = 1 := by
…
```
Mentions: Erdos829.cubes
theorem Erdos829.sumRep_cubes_two The only ordered pair of cubes summing to $2$ is $(1, 1)$ line 71 source
Source documentation
The only ordered pair of cubes summing to $2$ is $(1, 1)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, AdditiveCombinatorics.sumRep, Erdos829.cubes, OfNat.ofNat
Proof uses: Eq.mpr, Finset.card, Prod, Finset.filter, And, Exists, LT.lt, HAdd.hAdd, Prod.fst, HPow.hPow, Prod.snd, Finset.HasAntidiagonal.antidiagonal, id, congrFun', congrArg, Eq.trans, Membership.mem, Set, Classical.propDecidable, AdditiveCombinatorics.sumRep_def and 9 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem sumRep_cubes_two : sumRep cubes 2 = 1 := by
…
```
Mentions: Erdos829.cubes
theorem Erdos829.sumRep_cubes_three The integer $3$ is not the sum of two cubes line 77 source
Source documentation
The integer $3$ is not the sum of two cubes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, AdditiveCombinatorics.sumRep, Erdos829.cubes, OfNat.ofNat
Proof uses: Eq.mpr, Finset.card, Prod, Finset.filter, And, Exists, LT.lt, HAdd.hAdd, Prod.fst, HPow.hPow, Prod.snd, Finset.HasAntidiagonal.antidiagonal, id, congrFun', congrArg, Eq.trans, Membership.mem, Set, Classical.propDecidable, AdditiveCombinatorics.sumRep_def and 9 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem sumRep_cubes_three : sumRep cubes 3 = 0 := by
…
```
Mentions: Erdos829.cubes
theorem Erdos829.sumRep_cubes_taxicab The Hardy-Ramanujan taxicab number satisfies $1729 = 1^3 + 12^3 = 9^3 + 10^3$, giving the four ordered representations $(1, 1728), (1728, 1), (729, 1000), (1000, 729)$ line 84 source
Source documentation
The Hardy-Ramanujan taxicab number satisfies $1729 = 1^3 + 12^3 = 9^3 + 10^3$, giving the four ordered representations $(1, 1728), (1728, 1), (729, 1000), (1000, 729)$.

extra axioms: _private.0.Erdos829.sumRep_cubes_taxicab._native.decide.ax_1 what this means
Statement uses: Eq, Nat, AdditiveCombinatorics.sumRep, Erdos829.cubes, OfNat.ofNat
Proof uses: Eq.mpr, Finset.card, Prod, Finset.filter, And, Exists, LT.lt, HAdd.hAdd, Prod.fst, HPow.hPow, Prod.snd, Finset.HasAntidiagonal.antidiagonal, id, congrFun', congrArg, Eq.trans, Membership.mem, Set, Classical.propDecidable, AdditiveCombinatorics.sumRep_def and 5 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem sumRep_cubes_taxicab : sumRep cubes 1729 = 4 := by
…
```
Mentions: Erdos829.cubes
theorem Erdos829.variants.mordell Mordell proved $\limsup_{n \to \infty} (1_A \ast 1_A)(n) = \infty$, where $A$ is the set of perfect cubes sorry line 96 source
Source documentation
Mordell proved $\limsup_{n \to \infty} (1_A \ast 1_A)(n) = \infty$, where $A$ is the set of perfect cubes. Equivalently, the number of representations of $n$ as a sum of two cubes is unbounded.

depends on sorry what this means
Statement uses: Eq, ENat, Filter.limsup, Nat, AdditiveCombinatorics.sumRep, Erdos829.cubes, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem mordell : limsup (fun n : ℕ => (sumRep cubes n : ℕ∞)) atTop = (⊤ : ℕ∞) := by
…
```
Mentions: Erdos829.cubes
theorem Erdos829.variants.mahler Mahler proved $(1_A \ast 1_A)(n) \gg (\log n)^{1/4}$ for infinitely many $n$, where $A$ is the set of perfect cubes sorry line 107 source
Source documentation
Mahler proved $(1_A \ast 1_A)(n) \gg (\log n)^{1/4}$ for infinitely many $n$, where $A$ is the set of perfect cubes.
[Ma35b] Mahler, K., On the lattice points on curves of genus 1. Proc. London Math. Soc. (2) (1935), 431-466.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Frequently, Nat, LE.le, HMul.hMul, HPow.hPow, Real.log, HDiv.hDiv, AdditiveCombinatorics.sumRep, Erdos829.cubes, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem mahler : ∃ C > (0 : ℝ),
    ∃ᶠ (n : ℕ) in atTop, C * (Real.log n) ^ ((1 : ℝ) / 4) ≤ (sumRep cubes n : ℝ) := by
…
```
Mentions: Erdos829.cubes
theorem Erdos829.variants.stewart Stewart improved Mahler's lower bound to $(1_A \ast 1_A)(n) \gg (\log n)^{11/13}$ for infinitely many $n$, where $A$ is the set of perfect cubes sorry line 119 source
Source documentation
Stewart improved Mahler's lower bound to $(1_A \ast 1_A)(n) \gg (\log n)^{11/13}$ for infinitely many $n$, where $A$ is the set of perfect cubes.
[St08] Stewart, C. L., Cubic Thue equations with many solutions. Int. Math. Res. Not. IMRN (2008), Art. ID rnn040, 11.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Frequently, Nat, LE.le, HMul.hMul, HPow.hPow, Real.log, HDiv.hDiv, AdditiveCombinatorics.sumRep, Erdos829.cubes, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem stewart : ∃ C > (0 : ℝ),
    ∃ᶠ (n : ℕ) in atTop, C * (Real.log n) ^ ((11 : ℝ) / 13) ≤ (sumRep cubes n : ℝ) := by
…
```
Mentions: Erdos829.cubes