One three negative integer restriction
ProvedD1Bound5.one_three_negative_integer_restrictiondiophantine-tuplesnumber-theory
Fixed-field version of the negative common-extension restriction.
Preamble
import Mathlib.Algebra.Polynomial.Monic import Mathlib.Algebra.QuadraticAlgebra.Basic import Mathlib.Algebra.Ring.Basic import Mathlib.Algebra.Ring.Equiv import Mathlib.Algebra.Squarefree.Basic import Mathlib.Analysis.Complex.Norm import Mathlib.Data.Finset.Card import Mathlib.Data.Fintype.EquivFin import Mathlib.Data.Int.GCD 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
Formal statement
section
open scoped NumberField
open Module
namespace D1Bound5
theorem _root_.D1Bound5.one_three_negative_integer_restriction {K : Type*} [Field K]
[NumberField K] [NumberField.IsTotallyComplex K]
(d : ℤ) (hd : 0 < d)
(hparam : ∀ m : ℤ, m ≤ 0 → (∃ r : 𝓞 K, r*r = (m : 𝓞 K)) →
∃ y : ℤ, m = -d*y^2)
{n : ℤ} (hn : n < 0)
(hr : ∃ r : 𝓞 K, r*r = (n : 𝓞 K)+1)
(hs : ∃ s : 𝓞 K, s*s = 3*(n : 𝓞 K)+1) :
d=2 ∧ n%2=1 := by sorry
end D1Bound5
end
Source
D1Bound5/OneThreeNegative.lean, lines 110-124, SHA-256 608590903d9651b83081c7c7b14962c497bd29944ae7411568eb23eee5734f08. Project: https://github.com/tomislav-math/d1-bound5-lean