305.lean`

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

7 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 305

References:

erdosproblems.com/305
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[BlEr76] Bleicher, M. N. and Erdős, P., Denominators of unit fractions. J. Number Th. (1976), 157-168.
[Yo88] Yokota, H., On a problem of Bleicher and Erdős. J. Number Theory (1988), 198-207.
[LiSa24] Liu, Y. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113 (2024).

Imports (1)

FormalConjecturesUtil

def Erdos305.D D a b is the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ line 44 source
Source documentation
D a b is the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, And, Not, Membership.mem, OfNat.ofNat, LE.le, Eq, Rat, Finset.sum, HDiv.hDiv
11 instances, coercions and auxiliary constants left out.
```
noncomputable def D (a b : ℕ) : ℕ :=
…
```
def Erdos305.Dmax $D(b)=\max_{1\leq a<b}D(a,b)$ line 49 source
Source documentation
$D(b)=\max_{1\leq a<b}D(a,b)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Finset.Ico, OfNat.ofNat, Erdos305.D
6 instances, coercions and auxiliary constants left out.
```
noncomputable def Dmax (b : ℕ) : ℕ := (Finset.Ico 1 b).sup fun a ↦ D a b
…
```
Mentions: Erdos305.D
theorem Erdos305.erdos_305 sorry line 68 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, LT.lt, Filter.Eventually, LE.le, Erdos305.Dmax, HMul.hMul, HPow.hPow, Real.log, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
37 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos305.erdos_305.variants.bleicher_erdos Bleicher and Erdős [BlEr76] have shown that $D(b)\ll b(\log b)^2$ sorry line 75 source
Source documentation
Bleicher and Erdős [BlEr76] have shown that $D(b)\ll b(\log b)^2$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos305.Dmax, HMul.hMul, HPow.hPow, Real.log, Filter.atTop
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_305.variants.bleicher_erdos : ∃ C : ℝ, 0 < C ∧
    ∀ᶠ b : ℕ in atTop, (Dmax b : ℝ) ≤ C * b * (log b) ^ 2 := by
…
```
Mentions: Erdos305.Dmax
theorem Erdos305.erdos_305.variants.prime_lower_bound If $b=p$ is a prime then $D(p) \gg p\log p$ sorry line 81 source
Source documentation
If $b=p$ is a prime then $D(p) \gg p\log p$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Nat.Prime, LE.le, HMul.hMul, Real.log, Erdos305.Dmax, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_305.variants.prime_lower_bound : ∃ c : ℝ, 0 < c ∧
    ∀ᶠ p : ℕ in atTop, p.Prime → c * p * log p ≤ Dmax p := by
…
```
Mentions: c, Erdos305.Dmax
theorem Erdos305.erdos_305.variants.yokota Yokota [Yo88] proved that $D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2$ sorry line 89 source
Source documentation
Yokota [Yo88] proved that $D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos305.Dmax, HMul.hMul, Real.log, HPow.hPow, Filter.atTop
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_305.variants.yokota : ∃ C : ℝ, 0 < C ∧ ∀ᶠ b : ℕ in atTop,
    (Dmax b : ℝ) ≤ C * b * log b * log (log b) ^ 4 * log (log (log b)) ^ 2 := by
…
```
Mentions: Erdos305.Dmax
theorem Erdos305.erdos_305.variants.liu_sawhney Liu and Sawhney [LiSa24] proved that $D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}$ sorry line 97 source
Source documentation
Liu and Sawhney [LiSa24] proved that $D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos305.Dmax, HMul.hMul, Real.log, HPow.hPow, Filter.atTop
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_305.variants.liu_sawhney : ∃ C K : ℝ, 0 < C ∧ ∀ᶠ b : ℕ in atTop,
    (Dmax b : ℝ) ≤ C * b * log b * log (log b) ^ 3 * log (log (log b)) ^ K := by
…
```
Mentions: Erdos305.Dmax