859.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 859

Reference: erdosproblems.com/859

Imports (1)

FormalConjecturesUtil

def Erdos859.DivisorSumSet DivisorSumSet t is the set of natural numbers n such that t can be represented as a sum of distinct divisors of n line 34 source
Source documentation
DivisorSumSet t is the set of natural numbers n such that t can be represented as a sum of distinct divisors of n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Exists, Finset, And, LE.le, Nat.divisors, Eq, Finset.sum
4 instances, coercions and auxiliary constants left out.
```
def DivisorSumSet (t : ℕ) := { n : ℕ | ∃ s ⊆ Nat.divisors n, t = ∑ i ∈ s, i }
…
```
theorem Erdos859.erdos_859.variants.erdos_upper_lower_bounds Documented declaration: erdos_upper_lower_bounds sorry line 43 source
Source documentation
A weaker version of the problem proved by Erdos: The density dₜ of DivisorSumSet (t : ℕ) is bounded from below by 1 / log (t) ^ c₃ and from above by 1 / log (t) ^ c₄ for some positive constants c₃ and c₄.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Filter.Eventually, Set.HasDensity, Erdos859.DivisorSumSet, Set.univ, LT.lt, HDiv.hDiv, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
33 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_859.variants.erdos_upper_lower_bounds : ∃ᵉ (c₃ > (0 : ℝ)) (c₄ > (0 : ℝ)) (t₀ : ℕ),
    ∀ᶠ t in atTop, ∃ dₜ : ℝ, (DivisorSumSet t).HasDensity dₜ ∧
    1 / Real.log t ^ c₃ < dₜ ∧ dₜ < 1 / Real.log t ^ c₄ := by
…
```
Mentions: Erdos859.DivisorSumSet
theorem Erdos859.erdos_859 The density of the divisor sum set is asymptotically equivalent to $c_1 / \log(t)^{c_2}$ sorry line 56 source
Source documentation
The density of the divisor sum set is asymptotically equivalent to $c_1 / \log(t)^{c_2}$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Set.HasDensity, Erdos859.DivisorSumSet, Set.univ, Asymptotics.IsEquivalent, Filter.atTop, HDiv.hDiv, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_859 :
    ∃ c₁ > 0, ∃ c₂ > (0 : ℝ), ∃ d : ℕ → ℝ, (∀ t > 0, (DivisorSumSet t).HasDensity (d t)) ∧
      (fun (t : ℕ) ↦ d t) ~[atTop] (fun t ↦ c₁ / Real.log t ^ c₂) := by
…
```
Mentions: Erdos859.DivisorSumSet
lemma Erdos859.erdos_859.variants.trivial_case A case where we can easily calculate the density of DivisorSumSet t is that of t=0 line 65 source
Source documentation
A case where we can easily calculate the density of DivisorSumSet t is that of t=0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Set, Nat, Erdos859.DivisorSumSet, OfNat.ofNat, Set.univ
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Exists, Finset, And, LE.le, Nat.divisors, Finset.sum, Set.ofPred, funext, EmptyCollection, EmptyCollection.emptyCollection, eq_true, Exists.intro, congr, eq_self, and_self
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma erdos_859.variants.trivial_case : DivisorSumSet 0 = Set.univ := by
…
```
Mentions: Erdos859.DivisorSumSet
lemma Erdos859.erdos_859.variants.positive_density An easy sanity check is to prove that for every natural number t the density dₜ is a positive number sorry line 74 source
Source documentation
An easy sanity check is to prove that for every natural number t the density dₜ is a positive number. Hint: investigate some multiplicative structure of DivisorSumSet t.

depends on sorry what this means
Statement uses: Nat, Set.HasPosDensity, Erdos859.DivisorSumSet, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma erdos_859.variants.positive_density (t : ℕ) :
    (DivisorSumSet t).HasPosDensity := by
…
```
Mentions: Erdos859.DivisorSumSet