Three not square quad Int

Proved

D1Bound5.three_not_square_quadInt

by tp · Oct 24, 2026 · Mathlib 0df444a (Lean v4.33.1)

diophantine-tuplesnumber-theory

The rational integer three is not a square in any positive-discriminant quadratic integer coordinate ring, including all half-integral models.

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

Formal statement

section
open scoped NumberField
open Module

namespace D1Bound5
open QuadInt




theorem _root_.D1Bound5.three_not_square_quadInt {tau nu : ℤ}
    (hpos : 0 < discriminant tau nu) :
    ¬ ∃ r : QuadInt tau nu, r*r = 3 := by sorry









end D1Bound5
end

Source

D1Bound5/SmallQuadruple.lean, lines 23-32, SHA-256 711961f35418258108807ffc2f351fdfe390fbcae78fcb16809267ab419a0806. Project: https://github.com/tomislav-math/d1-bound5-lean

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