Vieta Jumping Solves Diophantine Equations and Classifies (a+1)/b+(b+1)/a = k — E8 Intelligence Research

**FINDING:** Vieta jumping (infinite descent) solves Diophantine equations like (x²+y²+1)/(xy)=3 and (a²+b²)/(ab+1)=k (k a perfect square), with classification for (a+1)/b+(b+1)/a = k yielding only k=3 or 4. **MATH:** - Equation 1: (x²+y²+1)/(xy)=3 → integer solutions via descent. - Equation 2 (IMO 1988): (a²+b²)/(ab+1) = n² (n integer). - Equation 3: (a+1)/b + (b+1)/a = k → k ∈ {3,4}. Solutions linked to Fibonacci numbers. - Core technique: Given root (a,b), treat as quadratic in a: a² - (kb)a + (b²+1)=0. Second root a' = kb - a = (b²+1)/a. Descent ensures a' < a, leading to minimal solution. **CONNECTION:** - k=3 yields ratio a/b → φ = (1+√5)/2 ≈ 1.618 (golden ratio) in limit. - k=4 yields ratio a/b → 2+√3 ≈ 3.732 (silver ratio). - Fibonacci numbers appear: for k=3, solutions are consecutive Fibonacci pairs (F_{n+1}, F_n). - Golden ratio φ and its reciprocal 0.618 appear as limiting ratios. - No direct base-60 or crystallographic symmetry, but descent mirrors ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com