218.lean`

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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 218

Reference: erdosproblems.com/218

Imports (1)

FormalConjecturesUtil

theorem Erdos218.erdos_218.variants.le The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a natural density of $\frac 1 2$ sorry line 35 source
Source documentation
The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a natural density of $\frac 1 2$.

depends on sorry what this means
Statement uses: Set.HasDensity, Nat, Set.ofPred, LE.le, primeGap, HAdd.hAdd, OfNat.ofNat, HDiv.hDiv, Real, Set.univ
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_218.variants.le : {n | primeGap n ≤ primeGap (n + 1)}.HasDensity <| 1 / 2 := by
…
```
Mentions: primeGap
theorem Erdos218.erdos_218.variants.ge The set of indices $n$ for which a prime gap is preceded by a larger or equal prime gap has a natural density of $\frac 1 2$ sorry line 43 source
Source documentation
The set of indices $n$ for which a prime gap is preceded by a larger or equal prime gap has a natural density of $\frac 1 2$.

depends on sorry what this means
Statement uses: Set.HasDensity, Nat, Set.ofPred, LE.le, primeGap, HAdd.hAdd, OfNat.ofNat, HDiv.hDiv, Real, Set.univ
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_218.variants.ge : {n | primeGap (n + 1) ≤ primeGap n}.HasDensity <| 1 / 2 := by
…
```
Mentions: primeGap
theorem Erdos218.erdos_218.variants.infinite_equal_prime_gap There are infinitely many indices $n$ such that the prime gap at $n$ is equal to the prime gap at $n+1$ sorry line 52 source
Source documentation
There are infinitely many indices $n$ such that the prime gap at $n$ is equal to the prime gap at $n+1$. This is equivalent to the existence of infinitely many arithmetic progressions of length $3$, see erdos_141.variants.infinite_three.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, Eq, primeGap, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_218.variants.infinite_equal_prime_gap : {n | primeGap n = primeGap (n + 1)}.Infinite := by
…
```
Mentions: primeGap