Vieta Jumping: Infinite Descent via Quadratic Symmetry in Diophantine Equations — E8 Intelligence Research

FINDING: Vieta jumping is a method of infinite descent used to solve Diophantine equations by exploiting symmetry in quadratic roots, with a canonical example being IMO 1988 Problem 6: if (a²+b²)/(ab+1) is an integer, then it is a perfect square. | MATH: For positive integers a,b, let k = (a²+b²)/(ab+1) ∈ ℕ. Fix k, treat as quadratic in a: a² - k b a + (b² - k) = 0. If (a,b) is a solution with a ≥ b, then the other root a' = k b - a is integer, positive, and smaller than b, enabling infinite descent until a=b, forcing k = a². | CONNECTION: The descent structure mirrors root system reflections in Lie algebras: the transformation (a,b) → (b, k b - a) is analogous to a Weyl group reflection in a rank-2 root system (e.g., A₂ or G₂). The integer condition k = (a²+b²)/(ab+1) being a square corresponds to the squared norm of a lattice vector in a scaled root lattice. The ratio a/b often approaches the golden ratio φ = (1+√5)/2 ≈ 1.618 in extremal cases, linking to the metallic means family. | Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com