674.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 674

References:

erdosproblems.com/674
[Ko40] Ko, Chao, Note on the Diophantine equation $x^xy^y=z^z$. J. Chinese Math. Soc. (1940), 31-39.

Imports (1)

FormalConjecturesUtil

def Erdos674.solutionSet The set of integer solutions to $x^x y^y = z^z$ with $x, y, z > 1$ line 34 source
Source documentation
The set of integer solutions to $x^x y^y = z^z$ with $x, y, z > 1$.

standard axioms: propext what this means
Statement uses: Set, Prod, Nat
Proof uses: Set.ofPred, Erdos674.solutionSet.match_1, And, LT.lt, OfNat.ofNat, Eq, HMul.hMul, HPow.hPow
8 instances, coercions and auxiliary constants left out.
```
def solutionSet : Set (ℕ × ℕ × ℕ) :=
…
```
theorem Erdos674.erdos_674 Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$? sorry line 44 source
Source documentation
Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$?
Ko [Ko40] proved there are none if $(x,y)=1$, but there are in fact infinitely many solutions in general - for example, $x=2^{12}3^6$, $y = 2^83^8$, and $z = 2^{11}3^7$.

depends on sorry what this means
Statement uses: Iff, True, Set.Nonempty, Prod, Nat, Erdos674.solutionSet
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_674 : answer(True) ↔ solutionSet.Nonempty := by
…
```
theorem Erdos674.erdos_674.variants.infinite There are in fact infinitely many integer solutions to $x^xy^y=z^z$ with $x,y,z>1$ sorry line 51 source
Source documentation
There are in fact infinitely many integer solutions to $x^xy^y=z^z$ with $x,y,z>1$.

depends on sorry what this means
Statement uses: Set.Infinite, Prod, Nat, Erdos674.solutionSet
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_674.variants.infinite : solutionSet.Infinite := by
…
```