723.lean`

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

5 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 723: The prime power conjecture.

Reference: erdosproblems.com/723

Imports (1)

FormalConjecturesUtil

theorem Erdos723.erdos_723 If there is a finite projective plane of order $n$ then must $n$ be a prime power? sorry line 36 source
Source documentation
If there is a finite projective plane of order $n$ then must $n$ be a prime power?

depends on sorry what this means
Statement uses: Iff, True, Membership, Fintype, Configuration.ProjectivePlane, IsPrimePow, Nat, Configuration.ProjectivePlane.order
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_723 :
    answer(sorry) ↔ ∀ {P L : Type} (_: Membership P L) (_ : Fintype P) (_ : Fintype L),
      ∀ pp : ProjectivePlane P L, IsPrimePow pp.order := by
…
```
theorem Erdos723.erdos_723.variants.prime_power_is_projplane_order These always exist if $n$ is a prime power sorry line 45 source
Source documentation
These always exist if $n$ is a prime power.

depends on sorry what this means
Statement uses: Nat, IsPrimePow, Exists, Membership, Fintype, Configuration.ProjectivePlane, Eq, Configuration.ProjectivePlane.order
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_723.variants.prime_power_is_projplane_order :
    ∀ n, IsPrimePow n → ∃ (P L : Type) (_ : Membership P L) (_ : Fintype P) (_ : Fintype L)
      (pp : ProjectivePlane P L), pp.order = n := by
…
```
theorem Erdos723.erdos_723.variants.leq_11 This conjecture has been proved for $n \leq 11$ sorry line 54 source
Source documentation
This conjecture has been proved for $n \leq 11$.

depends on sorry what this means
Statement uses: Membership, Fintype, Configuration.ProjectivePlane, LE.le, Nat, Configuration.ProjectivePlane.order, OfNat.ofNat, IsPrimePow
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_723.variants.leq_11 {P L : Type} [Membership P L] [Fintype P] [Fintype L] :
    ∀ pp : ProjectivePlane P L, pp.order ≤ 11 → IsPrimePow pp.order := by
…
```
theorem Erdos723.erdos_723.variants.eq_12 It is open whether there exists a projective plane of order 12 sorry line 62 source
Source documentation
It is open whether there exists a projective plane of order 12.

depends on sorry what this means
Statement uses: Iff, True, Exists, Membership, Fintype, Configuration.ProjectivePlane, Eq, Nat, Configuration.ProjectivePlane.order, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_723.variants.eq_12 : answer(sorry) ↔
    ∃ (P L : Type) (_ : Membership P L) (_ : Fintype P) (_ : Fintype L) (pp : ProjectivePlane P L),
      pp.order = 12 := by
…
```
theorem Erdos723.erdos_723.variants.bruck_ryser Bruck and Ryser have proved that if $n \equiv 1 (\mod 4)$ or $n \equiv 2 (\mod 4)$ then $n$ must be the sum of two squares sorry line 72 source
Source documentation
Bruck and Ryser have proved that if $n \equiv 1 (\mod 4)$ or $n \equiv 2 (\mod 4)$ then $n$ must be the sum of two squares.

depends on sorry what this means
Statement uses: Membership, Fintype, Nat, Configuration.ProjectivePlane, Eq, Configuration.ProjectivePlane.order, Or, Nat.ModEq, OfNat.ofNat, Exists, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_723.variants.bruck_ryser {P L : Type} [Membership P L] [Fintype P] [Fintype L]
    (n : ℕ) (pp : ProjectivePlane P L) (hpp : pp.order = n) :
    (n ≡ 1 [MOD 4] ∨ n ≡ 2 [MOD 4]) → ∃ a b, n = a ^ 2 + b ^ 2 := by
…
```