LegendreConjecture.lean`

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

3 declarations, 3 contain sorry, 0 declared as axioms

Legendre's conjecture

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/ErdosProblems/375.lean

theorem LegendreConjecture.legendre_conjecture Does there always exist at least one prime between consecutive perfect squares? sorry line 38 source
Source documentation
Does there always exist at least one prime between consecutive perfect squares?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Exists, And, Membership.mem, Set, Set.Ioo, HPow.hPow, HAdd.hAdd, Nat.Prime
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 legendre_conjecture :
    answer(sorry) ↔ ∀ n ≥ 1, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p := by
…
```
theorem LegendreConjecture.bounded_gap_legendre Documented declaration: bounded_gap_legendre sorry line 50 source
Source documentation
If there exists a constant c > 0 such that (n + 1).nth Nat.Prime - n.nth Nat.Prime < (n.nth Nat.Prime) ^ (1 / 2 - c) for all large n, then Legendre's conjecture is asymptotically true.
Formal proof linked here provided by AlphaProof.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LT.lt, HSub.hSub, Nat.nth, Nat.Prime, HAdd.hAdd, HPow.hPow, HDiv.hDiv, Filter.atTop, Membership.mem, Set, Set.Ioo
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem bounded_gap_legendre
    (H : ∃ c > 0, ∀ᶠ n in atTop, (n + 1).nth Nat.Prime - n.nth Nat.Prime <
      (n.nth Nat.Prime : ℝ) ^ (1 / (2 : ℝ) - c)) :
    ∀ᶠ n in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p := by
…
```
Mentions: c
theorem LegendreConjecture.legendre_conjecture.ferreira_large_n Ferreira proved that the conjecture is true for sufficiently large n sorry line 60 source
Source documentation
Ferreira proved that the conjecture is true for sufficiently large n.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Exists, And, Membership.mem, Set, Set.Ioo, HPow.hPow, OfNat.ofNat, HAdd.hAdd, Nat.Prime, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem legendre_conjecture.ferreira_large_n :
    ∀ᶠ n in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p := by
…
```