Fourth-element norm bound for case 5

Proved

D1Bound5.oneSmallFourth_5_growth

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

diophantine-tuplesnumber-theory

For the specified anchors a, b and s, every element z below the stated norm cutoff for which az + 1, bz + 1 and sz + 1 are squares satisfies N(z) ≤ 4N(b)N(s).

Preamble

import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.Data.List.Range
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Definitions.Def_D1Bound5_Coordinates_Definitions
import Definitions.Def_D1Bound5_OneSmallFourthData1_Definitions
import Definitions.Def_D1Bound5_OneSmallThirdData_Definitions_001

Formal statement

section
namespace D1Bound5
open QuadInt
set_option maxRecDepth 32768
set_option maxHeartbeats 12000000









































theorem _root_.D1Bound5.oneSmallFourth_5_growth (z : QuadInt 0 1)
    (hz : z.norm < oneSmallTripleCutoff_5)
    (ha : ∃ q, q*q = oneSmallPairA_2*z+1)
    (hb : ∃ q, q*q = oneSmallPairB_2*z+1)
    (hs : ∃ q, q*q = oneSmallTripleS_5*z+1) :
    z.norm ≤ 4*oneSmallPairB_2.norm*oneSmallTripleS_5.norm := by sorry















end D1Bound5
end

Source

D1Bound5/OneSmallFourthData1.lean, lines 116-126, SHA-256 fbe5d4af355a60f4a8c89a4d8ea6491dec3a75d17233e2ab145f623df38cf125. Project: https://github.com/tomislav-math/d1-bound5-lean

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