Degenerate pair model
ProvedD1Bound5.degenerate_pair_modeldiophantine-tuplesnumber-theory
The only possible nonrational distinct product-minus-one pair lies in the Eisenstein model. No smallness hypothesis is needed.
Preamble
import Mathlib.Algebra.QuadraticAlgebra.Basic import Mathlib.Algebra.Ring.Basic import Mathlib.Algebra.Ring.Equiv import Mathlib.Analysis.Complex.Norm import Mathlib.Data.Fin.Tuple.Sort import Mathlib.Data.Finset.Card import Mathlib.Data.Finset.Prod import Mathlib.Data.Fintype.EquivFin import Mathlib.Data.Int.GCD import Mathlib.Data.Int.Interval import Mathlib.Data.Int.Sqrt import Mathlib.LinearAlgebra.Dimension.Free import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex import Mathlib.NumberTheory.NumberField.Norm import Mathlib.Tactic.IntervalCases import Mathlib.Tactic.Linarith import Mathlib.Tactic.LinearCombination import Definitions.Def_D1Bound5_Coordinates_Definitions import Definitions.Def_D1Bound5_DegeneratePairClassification_Definitions
Formal statement
section
namespace D1Bound5
open QuadInt
theorem _root_.D1Bound5.degenerate_pair_model {tau nu : ℤ}
(htau : tau=0 ∨ tau=1) (hd : 0 < discriminant tau nu)
(a b : QuadInt tau nu) (hne : a ≠ b) (hprod : a*b+1=0) :
RationalUnitPair a b ∨ (tau=1 ∧ nu=1) := by sorry
end D1Bound5
end
Source
D1Bound5/DegeneratePairClassification.lean, lines 48-88, SHA-256 61da8c7aa871a81d9d56989ed28825c79c1cef687d7de61a56038260054cdd37. Project: https://github.com/tomislav-math/d1-bound5-lean