912.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 912

References:

erdosproblems.com/912
[Er82c] Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.

Imports (1)

FormalConjecturesUtil

def Erdos912.h If $n! = \prod_{i}p_i^{k_i}$ is the factorization into distinct primes, then we define $h(n)$ to be the number of distinct exponents $k_i$ line 37 source
Source documentation
If $n! = \prod_{i}p_i^{k_i}$ is the factorization into distinct primes, then we define $h(n)$ to be the number of distinct exponents $k_i$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finsupp.frange, Nat.factorization, Nat.factorial
2 instances, coercions and auxiliary constants left out.
```
noncomputable def h (n : ℕ) : ℕ := (n !).factorization.frange.card
…
```
theorem Erdos912.erdos_912.variants.selfridge Erdős and Selfridge prove in [Er82c] that $h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}$ sorry line 41 source
Source documentation
Erdős and Selfridge prove in [Er82c] that $h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}$.

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos912.h, HPow.hPow, HDiv.hDiv, Real.log, OfNat.ofNat
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_912.variants.selfridge :
    (fun n => (h n : ℝ)) =Θ[atTop] (fun n => (n / Real.log n) ^ (1 / 2 : ℝ)) := by
…
```
Mentions: Erdos912.h
theorem Erdos912.erdos_912 Prove that there exists some $c>0$ such that $$h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}$$ as $n\to \infty$ sorry line 50 source
Source documentation
Prove that there exists some $c>0$ such that $$h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}$$ as $n\to \infty$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsEquivalent, Nat, Filter.atTop, Erdos912.h, HMul.hMul, HPow.hPow, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_912 : ∃ c > 0,
    (fun n => (h n : ℝ)) ~[atTop] (fun n => c * (n / Real.log n) ^ (1 / 2 : ℝ)) := by
…
```
Mentions: c, Erdos912.h
theorem Erdos912.erdos_912.variants.tao A heuristic of Tao using the Cramér model for the primes suggests this is true with $c=\sqrt{2\pi}$ sorry line 57 source
Source documentation
A heuristic of Tao using the Cramér model for the primes suggests this is true with $c=\sqrt{2\pi}$.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos912.h, HMul.hMul, Real.sqrt, OfNat.ofNat, Real.pi, HPow.hPow, HDiv.hDiv, 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_912.variants.tao :
    (fun n => (h n : ℝ)) ~[atTop] (fun n => √(2 * Real.pi) * (n / Real.log n) ^ (1 / 2 : ℝ)) := by
…
```
Mentions: Erdos912.h