Pell Convergent Pair recurrence

Proved

D1Bound5.pellConvergentPair_recurrence

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

diophantine-tuplesnumber-theory

The forward recurrence that connects the short executable certificate to the independently proved real-convergent completeness theorem.

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
import Definitions.Def_D1Bound5_PellCFPrefixCheck_Definitions
import Definitions.Def_D1Bound5_PellConvergentCompleteness_Definitions

Formal statement

section
namespace D1Bound5




































theorem _root_.D1Bound5.pellConvergentPair_recurrence (x : ℝ) (n : ℕ) :
    pellConvergentPair x (n+2) =
      (⌊pellRealTail x (n+2)⌋ * (pellConvergentPair x (n+1)).1 +
          (pellConvergentPair x n).1,
       ⌊pellRealTail x (n+2)⌋ * (pellConvergentPair x (n+1)).2 +
          (pellConvergentPair x n).2) := by sorry















end D1Bound5
end

Source

D1Bound5/PellCFPrefixCheck.lean, lines 116-139, SHA-256 a5468d17d5b8fd8febaae9f486b606a6148827a88a3b76dca5782d036662adba. Project: https://github.com/tomislav-math/d1-bound5-lean

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