264.lean`

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6 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 264

Reference: erdosproblems.com/264

Imports (1)

FormalConjecturesUtil

def Erdos264.IsIrrationalitySequence Documented declaration: IsIrrationalitySequence line 45 source
Source documentation
A sequence $a_n$ of integers is called an irrationality sequence if for every bounded sequence of integers $b_n$ with $a_n + b_n \neq 0$ and $b_n \neq 0$ for all $n$, the sum $$ \sum \frac{1}{a_n + b_n} $$ is irrational.
Note: there are other possible definitions of this concept. See FormalConjectures/ErdosProblems/263.lean for another possible definition.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int, BddAbove, Set.range, BddBelow, Not, Membership.mem, Set, HAdd.hAdd, OfNat.ofNat, Irrational, tsum, Real, HDiv.hDiv, SummationFilter.unconditional
19 instances, coercions and auxiliary constants left out.
```
def IsIrrationalitySequence (a : ℕ → ℕ) : Prop := ∀ b : ℕ → ℤ,
…
```
theorem Erdos264.erdos_264.parts.i Is $2^n$ an example of an irrationality sequence? sorry line 56 source
Source documentation
Is $2^n$ an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24]
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

depends on sorry what this means
Statement uses: Not, Erdos264.IsIrrationalitySequence, Nat, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_264.parts.i : ¬IsIrrationalitySequence (2 ^ ·) := by sorry
…
```
Mentions: Erdos264.IsIrrationalitySequence
theorem Erdos264.erdos_264.parts.ii Is $n!$ an example of an irrationality sequence? sorry line 62 source
Source documentation
Is $n!$ an example of an irrationality sequence?

depends on sorry what this means
Statement uses: Iff, True, Erdos264.IsIrrationalitySequence, Nat.factorial
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_264.parts.ii : answer(sorry) ↔ IsIrrationalitySequence Nat.factorial := by sorry
…
```
Mentions: Erdos264.IsIrrationalitySequence
theorem Erdos264.erdos_264.variants.example One example is $2^{2^n}$ sorry line 68 source
Source documentation
One example is $2^{2^n}$.

depends on sorry what this means
Statement uses: Erdos264.IsIrrationalitySequence, Nat, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_264.variants.example : IsIrrationalitySequence (fun n ↦ 2 ^ (2 ^ n)) := by sorry
…
```
Mentions: Erdos264.IsIrrationalitySequence
theorem Erdos264.erdos_264.variants.ko_tao_neg Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers sorry line 81 source
Source documentation
Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers $a_n$ such that $\sum \frac{1}{a_n}$ converges and $$ \liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0 $$ is not an irrationality sequence.
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

depends on sorry what this means
Statement uses: Nat, StrictMono, Not, Membership.mem, Set, Set.range, OfNat.ofNat, Summable, Real, HDiv.hDiv, SummationFilter.unconditional, Exists, And, GT.gt, Filter.Eventually, LE.le, HMul.hMul, HPow.hPow, tsum, Set.Ioi and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_264.variants.ko_tao_neg {a : ℕ → ℕ} (h₁ : StrictMono a) (h₂ : 0 ∉ Set.range a)
    (h₃ : Summable ((1 : ℝ) / a ·))
    (h₄ : ∃ C > 0, ∀ᶠ n in atTop, C ≤ a n ^ 2 * ∑' k : Set.Ioi n, (1 : ℝ) / a k ^ 2) :
    ¬IsIrrationalitySequence a := by
…
```
Mentions: Erdos264.IsIrrationalitySequence
theorem Erdos264.erdos_264.variants.ko_tao_pos On the other hand, Kovač and Tao [KoTa24] do prove that for any function sorry line 94 source
Source documentation
On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ with $\lim_{n \to \infty} \frac{F(n + 1)}{F(n)} = \infty$ there exists such an irrationality sequence with $a_n \sim F(n)$.
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

depends on sorry what this means
Statement uses: Nat, Filter.Tendsto, Real, HDiv.hDiv, HAdd.hAdd, OfNat.ofNat, Filter.atTop, Exists, And, Erdos264.IsIrrationalitySequence, Asymptotics.IsEquivalent
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_264.variants.ko_tao_pos {F : ℕ → ℕ}
    (hF : atTop.Tendsto (fun n ↦ (F (n + 1) : ℝ) / F n) atTop) :
    ∃ a : ℕ → ℕ, IsIrrationalitySequence a ∧ (fun n ↦ (a n : ℝ)) ~[atTop] fun n ↦ (F n : ℝ) := by
…
```
Mentions: Erdos264.IsIrrationalitySequence