1099.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 1099

References:

erdosproblems.com/1099
[Er81h] Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.
[Vo84] Vose, Michael D., Integers with consecutive divisors in small ratio. J. Number Theory (1984), 233-238.
[Te87] Tenenbaum, Gérald, Sur un problème extrémal en arithmétique. Ann. Inst. Fourier (1987), 1-18.

Imports (1)

FormalConjecturesUtil

def Erdos1099.h For $\alpha>1$ and the divisors $1=d_1<\cdots<d_{\tau(n)}=n$ of $n$, $$h_\alpha(n) = \sum_i \left( \frac{d_{i+1}}{d_i}-1\right)^\alpha.$$ line 43 source
Source documentation
For $\alpha>1$ and the divisors $1=d_1<\cdots<d_{\tau(n)}=n$ of $n$, $$h_\alpha(n) = \sum_i \left( \frac{d_{i+1}}{d_i}-1\right)^\alpha.$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: Finset.sum, Fin, HSub.hSub, Finset.card, Nat.divisors, OfNat.ofNat, Finset.univ, HPow.hPow, HDiv.hDiv, Nat.nth, Dvd.dvd, HAdd.hAdd
19 instances, coercions and auxiliary constants left out.
```
noncomputable def h (α : ℝ) (n : ℕ) : ℝ :=
…
```
theorem Erdos1099.erdos_1099 sorry line 65 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Filter.Frequently, Nat, LE.le, Erdos1099.h, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1099.erdos_1099.variants.factorial Is $h_\alpha(n!)$ bounded? sorry line 72 source
Source documentation
Is $h_\alpha(n!)$ bounded? Tenenbaum [Te87, Théorème 1] proved that it is, for every $\alpha > 1$.

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Nat, LE.le, Erdos1099.h, Nat.factorial
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1099.variants.factorial : answer(True) ↔ ∀ α : ℝ, 1 < α →
    ∃ C : ℝ, ∀ n : ℕ, h α n.factorial ≤ C := by
…
```
Mentions: Erdos1099.h
theorem Erdos1099.erdos_1099.variants.lcm Is $h_\alpha(\mathrm{lcm}(1,\ldots,n))$ bounded? sorry line 79 source
Source documentation
Is $h_\alpha(\mathrm{lcm}(1,\ldots,n))$ bounded? Tenenbaum [Te87, Théorème 1] proved that it is, for every $\alpha > 1$.

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Nat, LE.le, Erdos1099.h, Finset.lcm, Finset.Icc, id
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1099.variants.lcm : answer(True) ↔ ∀ α : ℝ, 1 < α →
    ∃ C : ℝ, ∀ n : ℕ, h α ((Finset.Icc 1 n).lcm id) ≤ C := by
…
```
Mentions: Erdos1099.h
theorem Erdos1099.erdos_1099.variants.ratio_sum Erdős remarks that this problem occurred to him when considering $\sum_i \frac{d_{i+1}}{d_i}$ sorry line 90 source
Source documentation
Erdős remarks that this problem occurred to him when considering $\sum_i \frac{d_{i+1}}{d_i}$. It is easy to see that $\sum_{i} \frac{d_{i+1}}{d_i}> \tau(n)+\log n$, and Erdős asked whether $$\liminf \left(\sum_{i} \frac{d_{i+1}}{d_i}-\tau(n)-\log n\right)<\infty,$$ which follows from the affirmative answer to the main question.

depends on sorry what this means
Statement uses: Exists, Real, Filter.Frequently, Nat, LE.le, HSub.hSub, Finset.sum, Fin, Finset.card, Nat.divisors, OfNat.ofNat, Finset.univ, HDiv.hDiv, Nat.nth, Dvd.dvd, HAdd.hAdd, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1099.variants.ratio_sum : ∃ C : ℝ, ∃ᶠ n : ℕ in atTop,
    ∑ i : Fin (n.divisors.card - 1), ((Nat.nth (· ∣ n) (i + 1) : ℝ) / Nat.nth (· ∣ n) i) -
      n.divisors.card - Real.log n ≤ C := by
…
```