428.lean`

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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 428

Reference: erdosproblems.com/428

Imports (1)

FormalConjecturesUtil

def Erdos428.primeDensityRatio The density ratio of set $A$ up to $n$ relative to the prime counting function $\pi(n)$ line 35 source
Source documentation
The density ratio of set $A$ up to $n$ relative to the prime counting function $\pi(n)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Real
Proof uses: HDiv.hDiv, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Nat.primeCounting
8 instances, coercions and auxiliary constants left out.
```
noncomputable def primeDensityRatio (A : Set ℕ) (n : ℕ) : ℝ :=
…
```
theorem Erdos428.erdos_428 Is there a set sorry line 43 source
Source documentation
Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $$\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$$

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Filter.Frequently, Membership.mem, LT.lt, OfNat.ofNat, Nat.Prime, HSub.hSub, Filter.atTop, GT.gt, Real, Filter.liminf, Erdos428.primeDensityRatio
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_428 :
    answer(sorry) ↔ ∃ A : Set ℕ,
      (∃ᶠ n in atTop, ∀ a ∈ A, 0 < a → a < n → (n - a).Prime) ∧
      liminf (fun n ↦ primeDensityRatio A n) atTop > 0 := by
…
```
Mentions: Erdos428.primeDensityRatio