Vieta Jumping Classifies Integer Solutions to (a+1)/b + (b+1)/a = k — E8 Intelligence Research

FINDING: Complete classification of positive integer solutions to (a+1)/b + (b+1)/a = k via Vieta jumping, yielding k ∈ {3,4} and solution pairs linked to Fibonacci recurrence. MATH: Equation: (a+1)/b + (b+1)/a = k, k integer. Vieta jumping transforms (a,b) → (b, kb - a - 1). For k=4, solutions satisfy recurrence a_{n+1} = 4a_n - a_{n-1} - 1, with minimal solution (1,1). For k=3, recurrence a_{n+1} = 3a_n - a_{n-1} - 1, minimal (1,2). Cassini identity F_{n-1}F_{n+1} - F_n^2 = (-1)^n appears in linear Diophantine contexts. CONNECTION: Recurrence for k=4: a_{n+1} = 4a_n - a_{n-1} - 1 yields ratios approaching 2+√3 ≈ 3.732 (no golden ratio). For k=3: ratio approaches (3+√5)/2 ≈ 2.618 — this is φ² = (1.618)² = 2.618, a direct geometric harmony constant. The minimal solution (1,2) yields Fibonacci-like growth. No base-60 or crystallographic symmetry directly observed. DEPTH: 7 — Elegant classification using Vieta jumping reveals that only two integer k values are possible, and the k=3 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com