Pell CFSolutions complete abs

Proved

D1Bound5.pellCFSolutions_complete_abs

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

diophantine-tuplesnumber-theory

This is the cutoff used for the signed rational extensions: their magnitude is u^2-1, and the bound is X^2. Both Pell signs are retained via absolute values; no positivity assumption on the supplied roots is made.

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

Formal statement

section
namespace D1Bound5


















































theorem _root_.D1Bound5.pellCFSolutions_complete_abs {d a0 X u v : ℤ} {N : ℕ}
    (hd : 7 ≤ d) (hcheck : pellCFPrefixCheck d a0 N = true)
    (hstop : X < (pellCFAt d a0 (pellCFInit a0) N).p)
    (hX : 1 ≤ X) (hv : v ≠ 0) (heq : u^2-d*v^2=2)
    (hbound : u^2-1 ≤ X^2) :
    (|u|,|v|) ∈ pellCFSolutions d a0 X N := by sorry

end D1Bound5
end

Source

D1Bound5/PellCFPrefixCheck.lean, lines 296-317, SHA-256 a5468d17d5b8fd8febaae9f486b606a6148827a88a3b76dca5782d036662adba. Project: https://github.com/tomislav-math/d1-bound5-lean

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