Eisenstein exceptional pair extension trace zero

Proved

D1Bound5.eisenstein_exceptional_pair_extension_trace_zero

by tp · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

diophantine-tuplesnumber-theory

Every common extension of the exceptional Eisenstein unit pair (-ω,1-ω), whose product is -1, has zero trace. No bound on the extension or on its square roots is assumed.

Preamble

import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Definitions.Def_D1Bound5_Coordinates_Definitions

Formal statement

section
namespace D1Bound5
open QuadInt


theorem _root_.D1Bound5.eisenstein_exceptional_pair_extension_trace_zero
    (z : QuadInt 1 1)
    (hr : ∃ r : QuadInt 1 1, r*r = (⟨0,-1⟩ : QuadInt 1 1)*z+1)
    (hs : ∃ s : QuadInt 1 1, s*s = (⟨1,-1⟩ : QuadInt 1 1)*z+1) :
    2*z.re+z.im = 0 := by sorry

end D1Bound5
end

Source

D1Bound5/EisensteinExceptionalPair.lean, lines 6-34, SHA-256 fec9949d6b0e083b0e0e0a45fd489ab00443fa8d7608258ece4a624254276af4. Project: https://github.com/tomislav-math/d1-bound5-lean

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