FermatCatalanConjecture.lean`

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

5 declarations, 2 contain sorry, 0 declared as axioms

Fermat-Catalan conjecture

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

def FermatCatalanConjecture.FermatCatalanSet' The set of solutions to the Fermat-Catalan Conjecture, i.e. the set of solutions $(a,b,c,m,n,k)$ to the equation $a^m + b^n = c^k$ where $\frac 1 m + \frac 1 n + \frac 1 k < 1$ line 37 source
Source documentation
The set of solutions to the Fermat-Catalan Conjecture, i.e. the set of solutions $(a,b,c,m,n,k)$ to the equation $a^m + b^n = c^k$ where $\frac 1 m + \frac 1 n + \frac 1 k < 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Fin, OfNat.ofNat, Nat
Proof uses: Set.ofPred, And, LT.lt, Set.Pairwise, Insert.insert, Singleton.singleton, Function.onFun, Nat.Coprime, Eq, HAdd.hAdd, HPow.hPow, Real, Finset.sum, Finset.Icc, HDiv.hDiv
25 instances, coercions and auxiliary constants left out.
```
def FermatCatalanSet' : Set (Fin 6 → ℕ) :=
…
```
def FermatCatalanConjecture.FermatCatalanSet line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Prod, Nat
Proof uses: Set.image, Fin, OfNat.ofNat, Prod.mk, HPow.hPow, FermatCatalanConjecture.FermatCatalanSet'
7 instances, coercions and auxiliary constants left out.
def FermatCatalanConjecture.fermatCatalanConjecture The proposition that the Fermat-Catalan Conjecture is true line 48 source
Source documentation
The proposition that the Fermat-Catalan Conjecture is true.

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Set.Finite, Prod, Nat, FermatCatalanConjecture.FermatCatalanSet
```
def fermatCatalanConjecture : Prop :=
…
```
theorem FermatCatalanConjecture.fermat_catalan The Fermat–Catalan conjecture states that the equation sorry line 59 source
Source documentation
The Fermat–Catalan conjecture states that the equation $a^m + b^n = c^k$ has only finitely many solutions $(a,b,c,m,n,k)$ with distinct triplets of values $(a^m, b^n, c^k)$ where $a, b, c$ are positive coprime integers and $m, n, k$ are positive integers satisfying $\frac 1 m + \frac 1 n + \frac 1 k < 1$.

depends on sorry what this means
Statement uses: FermatCatalanConjecture.fermatCatalanConjecture
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem fermat_catalan : fermatCatalanConjecture := by
…
```
Mentions: FermatCatalanConjecture.fermatCatalanConjecture
theorem FermatCatalanConjecture.fermat_catalan.variants.darmon_granville By the Darmon-Granville theorem, for any fixed choice of positive integers m, n and k satisfying sorry line 68 source
Source documentation
By the Darmon-Granville theorem, for any fixed choice of positive integers m, n and k satisfying $\frac 1 m + \frac 1 n + \frac 1 k < 1$, only finitely many coprime triples $(a, b, c)$ solving $a^m + b^n = c^k$ exist.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Real, HAdd.hAdd, HDiv.hDiv, Set.Finite, Prod, Set.ofPred, FermatCatalanConjecture.fermat_catalan.variants.darmon_granville.match_1, And, Eq, HPow.hPow, Set.Pairwise, Insert.insert, Set, Singleton.singleton, Nat.Coprime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem fermat_catalan.variants.darmon_granville
    (m n k : ℕ) (hm : 0 < m) (hn : 0 < n) (hk : 0 < k)
    (H : (1 / m : ℝ) + 1 / n + 1 / k < 1) :
    {(a, b, c) : ℕ × ℕ × ℕ | 0 < a ∧ 0 < b ∧ 0 < c ∧ a^m + b^n = c^k ∧
      ({a, b, c} : Set _).Pairwise Nat.Coprime}.Finite := by
…
```
Mentions: c