Disjoint coordinate branches of the Hurwitz integers

Proved

HurwitzQ.not_allIntCoordsQ_and_allHalfIntCoordsQ

by jawneeboy · Sep 18, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebrahurwitz-integersnumber-theoryquaternions

Let $\mathcal{H}$ be the Hurwitz subring of the rational quaternions: the four coordinates of an element are either all in $\mathbb{Z}$ or all in $\mathbb{Z}+\frac12$ . Let $q$ be a rational quaternion.

\neg\bigl((a,b,c,d\in\mathbb{Z})\land(a,b,c,d\in\mathbb{Z}+\tfrac12)\bigr),\qquad q=a+bi+cj+dk.

The two alternatives in the Hurwitz coordinate criterion cannot hold simultaneously.

Preamble

import Definitions.Def_HurwitzQ_hurwitzIntegersQ
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring

open Quaternion QuaternionAlgebra HurwitzQ

Formal statement

theorem HurwitzQ.not_allIntCoordsQ_and_allHalfIntCoordsQ (q : ℍ[ℚ]) :
    ¬ (allIntCoordsQ q ∧ allHalfIntCoordsQ q) := by sorry

Source

Standard definition: John H. Conway and Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A K Peters, 2003, §5.1, The Hurwitz Integral Quaternions. https://www.routledge.com/On-Quaternions-and-Octonions/Conway-Smith/p/book/9781568811345 The displayed assertion is an elementary consequence of this definition; no numbered theorem attribution is claimed.

View graph