Oppermann.lean`

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

6 declarations, 4 contain sorry, 0 declared as axioms

Oppermann's Conjecture

References:

Imports (1)

FormalConjecturesUtil

theorem Oppermann.oppermann_conjecture.parts.i For every integer $x \ge 2$ there exists a prime between $x(x-1)$ and $x^2$ sorry line 38 source
Source documentation
For every integer $x \ge 2$ there exists a prime between $x(x-1)$ and $x^2$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, And, Membership.mem, Finset, Finset.Ioo, HMul.hMul, HSub.hSub, HPow.hPow, Nat.Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem oppermann_conjecture.parts.i (x : ℕ) (hx : 2 ≤ x) :
    ∃ p ∈ Ioo (x * (x - 1)) (x^2), p.Prime := by
…
```
theorem Oppermann.oppermann_conjecture.parts.ii For every integer $x \ge 2$ there exists a prime between $x^2$ and $x(x+1)$ sorry line 46 source
Source documentation
For every integer $x \ge 2$ there exists a prime between $x^2$ and $x(x+1)$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, And, Membership.mem, Finset, Finset.Ioo, HPow.hPow, HMul.hMul, HAdd.hAdd, Nat.Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem oppermann_conjecture.parts.ii (x : ℕ) (hx : 2 ≤ x) :
    ∃ p ∈ Ioo (x^2) (x * (x + 1)), p.Prime := by
…
```
theorem Oppermann.oppermann_conjecture Oppermann's Conjecture: For every integer $x \ge 2$, the following hold: - There exists a prime between $x(x-1)$ and $x^2$ sorry line 57 source
Source documentation
Oppermann's Conjecture: For every integer $x \ge 2$, the following hold:
- There exists a prime between $x(x-1)$ and $x^2$.
- There exists a prime between $x^2$ and $x(x+1)$.
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, Exists, Membership.mem, Finset, Finset.Ioo, HMul.hMul, HSub.hSub, HPow.hPow, Nat.Prime, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem oppermann_conjecture (x : ℕ) (hx : 2 ≤ x) :
    (∃ p ∈ Ioo (x * (x - 1)) (x^2), p.Prime) ∧
    (∃ p ∈ Ioo (x^2) (x * (x + 1)), p.Prime) := by
…
```
theorem Oppermann.oppermann_implies_brocard Oppermann's conjecture implies Brocard's conjecture line 64 source
Source documentation
Oppermann's conjecture implies Brocard's conjecture.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, Exists, Membership.mem, Finset, Finset.Ioo, HMul.hMul, HSub.hSub, HPow.hPow, Nat.Prime, HAdd.hAdd, Finset.card, Finset.filter, Nat.nth
Proof uses: Eq.mpr, id, Eq, congrArg, rfl, Nat.prime_nth_prime, Eq.rec, Iff.mpr, Nat.nth_le_nth, Nat.infinite_setOfPred_prime, Nat.nth_prime_one_eq_three, LT.lt, Nat.nth_strictMono, Nat.lt_succ_self, Odd, Nat.Prime.odd_of_ne_two, Decidable.byContradiction, Not, And.left, Nat.pow_le_pow_left and 101 more
125 instances, coercions and auxiliary constants left out.

Tactic trace: 42 steps
Open to load the checked before-and-after goal states.
```
theorem oppermann_implies_brocard (n : ℕ) (hn : 1 ≤ n) (P : type_of% oppermann_conjecture) :
    letI prev := n.nth Nat.Prime
…
```
Mentions: Oppermann.oppermann_conjecture
theorem Oppermann.oppermann_implies_legendre Oppermann's conjecture implies Legendre's conjecture line 106 source
Source documentation
Oppermann's conjecture implies Legendre's conjecture.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, Exists, Membership.mem, Finset, Finset.Ioo, HMul.hMul, HSub.hSub, HPow.hPow, Nat.Prime, HAdd.hAdd
Proof uses: Eq.mpr, id, Eq, Nat.Simproc.le_add_ge, of_decide_eq_true, Eq.refl, Bool, Bool.true, Exists.intro, And.intro, Finset.Ioo_subset_Ioo_left, of_eq_true, Eq.trans, True, congr, congrArg, sq, Nat.Simproc.add_sub_le, add_one_mul, And.left and 1 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem oppermann_implies_legendre (n : ℕ) (hn : 1 ≤ n) (P : type_of% oppermann_conjecture) :
    ∃ p ∈ Ioo (n ^ 2) ((n + 1) ^ 2), p.Prime := by
…
```
Mentions: Oppermann.oppermann_conjecture
theorem Oppermann.oppermann_conjecture.ferreira_large_x Ferreira proved that Oppermann's conjecture is true for sufficiently large x sorry line 117 source
Source documentation
Ferreira proved that Oppermann's conjecture is true for sufficiently large x.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, And, Exists, Membership.mem, Finset, Finset.Ioo, HMul.hMul, HSub.hSub, OfNat.ofNat, HPow.hPow, Nat.Prime, HAdd.hAdd, Filter.atTop
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 oppermann_conjecture.ferreira_large_x : ∀ᶠ x in atTop,
    (∃ p ∈ Ioo (x * (x - 1)) (x^2), p.Prime) ∧
    (∃ p ∈ Ioo (x^2) (x * (x + 1)), p.Prime) := by
…
```