Pell norm two is convergent

Proved

D1Bound5.pell_norm_two_is_convergent

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

diophantine-tuplesnumber-theory

The norm-two equation gives the strict Legendre approximation, including the reduced denominator. This is an actual Pell theorem, not an assumed continued-fraction enumeration.

Preamble

import Mathlib.Data.Rat.Lemmas
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic

Formal statement

section
namespace D1Bound5




















theorem _root_.D1Bound5.pell_norm_two_is_convergent {d x y : ℤ} (hd : 7 ≤ d)
    (hx : 0 < x) (hy : 0 < y) (heq : x^2-d*y^2=2) :
    ∃ n, (x/y : ℚ) = (Real.sqrt (d : ℝ)).convergent n := by sorry



end D1Bound5
end

Source

D1Bound5/PellConvergentCompleteness.lean, lines 119-154, SHA-256 c43074daa228222fbe1ca798070304a8be11d05710c382ec9a9e73d7799c8266. Project: https://github.com/tomislav-math/d1-bound5-lean

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