873.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 873

References:

erdosproblems.com/873
[Er92c] Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. 15 (1992), 34-50.

Imports (1)

FormalConjecturesUtil

abbrev Erdos873.F Let $a$ be some sequence of natural numbers line 36 source
Source documentation
Let $a$ be some sequence of natural numbers. We set $F(A,X,k)$ to be the count of the number of $i$ such that $[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X$, where the left-hand side is the least common multiple.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, ENat
Proof uses: Set.encard, Set.ofPred, LT.lt, Finset.lcm, Finset.range, HAdd.hAdd
8 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev F (a : ℕ → ℕ) (X : ℝ) (k : ℕ) : ℕ∞ :=
…
```
theorem Erdos873.erdos_873 Documented declaration: erdos_873 sorry line 44 source
Source documentation
Let $A = \{a_1 < a_2 < \dots\} \subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that $[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X$, where the left-hand side is the least common multiple. Is it true that, for every $\epsilon > 0$, there exists some $k$ such that $F(A,X,k) < X^\epsilon$?

depends on sorry what this means
Statement uses: Iff, True, Nat, Real, GT.gt, OfNat.ofNat, LT.lt, StrictMono, Exists, EReal, Erdos873.F, HPow.hPow
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_873 : answer(sorry) ↔ ∀ᵉ (a : ℕ → ℕ) (ε > (0 : ℝ)), 0 < a 0 → StrictMono a →
    ∃ k, ∀ X > 0, F a X k < (X^ε).toEReal := by
…
```
Mentions: Erdos873.F
theorem Erdos873.erdos_873.variants.triple_upper_bound The upper bound (2) sorry line 56 source
Source documentation
The upper bound (2).

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, StrictMono, Filter.Eventually, LE.le, EReal, Erdos873.F, HMul.hMul, HPow.hPow, HDiv.hDiv, Real.log, Filter.atTop
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_873.variants.triple_upper_bound :
    ∃ C : ℝ, 0 < C ∧
      ∀ (a : ℕ → ℕ), 0 < a 0 → StrictMono a →
        ∀ᶠ X : ℝ in Filter.atTop,
          (F a X 3 : EReal) ≤
            (C * X ^ (1 / 3 : ℝ) * Real.log X).toEReal := by
…
```
Mentions: Erdos873.F
theorem Erdos873.erdos_873.variants.triple_lower_bound_infinitely_often The infinitely-often lower bound (3) sorry line 66 source
Source documentation
The infinitely-often lower bound (3).

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, LT.lt, OfNat.ofNat, StrictMono, GT.gt, LE.le, EReal, HMul.hMul, HPow.hPow, HDiv.hDiv, Real.log, Erdos873.F
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_873.variants.triple_lower_bound_infinitely_often :
    ∃ (a : ℕ → ℕ) (c : ℝ),
      0 < a 0 ∧ StrictMono a ∧ 0 < c ∧
        ∀ X₀ : ℝ, ∃ X > X₀,
          (c * X ^ (1 / 3 : ℝ) * Real.log X).toEReal ≤ (F a X 3 : EReal) := by
…
```
Mentions: c, Erdos873.F
theorem Erdos873.erdos_873.variants.supplement_all_scale sorry line 79 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, Real, And, LT.lt, OfNat.ofNat, StrictMono, Filter.Eventually, LE.le, EReal, HMul.hMul, HPow.hPow, HDiv.hDiv, Real.log, Erdos873.F, Filter.atTop
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.