Vieta Jumping Reveals Fibonacci-Like Sequences in Quadratic Diophantine Equations — E8 Intelligence Research

FINDING: Vieta jumping reduces infinite descent in quadratic Diophantine equations to recurrence relations, often yielding Fibonacci-like sequences. | MATH: For equation (a² + b² + 5)/(ab) = 5, Vieta jumping gives recurrence (a, b) → (b, 5b - a). General form: if (x, y) solves (x² + y² + c)/(xy) = k, then (y, ky - x) is also a solution. | CONNECTION: Recurrence ratio tends to φ = 1.618... for large k; descent steps mirror golden ratio scaling. | DEPTH: 8 — Bridges quadratic forms, number theory, and harmonic sequences; reveals hidden symmetry in integer solutions. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com