932.lean`

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2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 932

Reference: erdosproblems.com/932

Imports (1)

FormalConjecturesUtil

theorem Erdos932.erdos_932 Let $p_k$ denote the $k$th prime sorry line 35 source
Source documentation
Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r < n < p_{r+1}$ all of whose prime factors are $< p_{r + 1} - p_r$.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, LE.le, OfNat.ofNat, Finset.card, Finset.filter, LT.lt, Nat.maxPrimeFac, HSub.hSub, Nat.nth, Nat.Prime, Nat.succ, Finset.Ioo
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_932 :
    { r : ℕ | 2 ≤ (Finset.Ioo (r.nth Nat.Prime) (r.succ.nth Nat.Prime) |>.filter
      (fun m => m.maxPrimeFac < r.succ.nth Nat.Prime - r.nth Nat.Prime)).card }.Infinite := by
…
```
theorem Erdos932.erdos_932.variants.one_le Erdős could show that the density of $r$ such that at least one such $n$ exists is $0$ sorry line 44 source
Source documentation
Erdős could show that the density of $r$ such that at least one such $n$ exists is $0$.

depends on sorry what this means
Statement uses: Set.HasDensity, Nat, Set.ofPred, LE.le, OfNat.ofNat, Finset.card, Finset.filter, LT.lt, Nat.maxPrimeFac, HSub.hSub, Nat.nth, Nat.Prime, Nat.succ, Finset.Ioo, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_932.variants.one_le :
    { r : ℕ | 1 ≤ (Finset.Ioo (r.nth Nat.Prime) (r.succ.nth Nat.Prime) |>.filter
      (fun m => m.maxPrimeFac < r.succ.nth Nat.Prime - r.nth Nat.Prime)).card }.HasDensity 0 := by
…
```