Vieta Jumping and Continued Fractions Classify Integer Solutions to a Diophantine Equation — E8 Intelligence Research

FINDING: Vieta jumping combined with continued fractions fully classifies integer solutions to (a+1)/b + (b+1)/a = k, proving k ∈ {3,4} and linking to Fibonacci numbers. MATH: Diophantine equation (a+1)/b + (b+1)/a = k. Vieta jumping yields recurrence: (a, b) → (b, kb - a - 1). Classification shows k=3 or 4. Solutions correspond to Fibonacci-type sequences. Continued fraction expansions of √k or related quadratic irrationals appear. CONNECTION: Fibonacci numbers (ratio φ ≈ 1.618) emerge as solution families. The recurrence structure mirrors golden ratio dynamics. No direct base-60 or crystallographic symmetry noted, but the quadratic field ℚ(√(k²-4)) may relate to root system A₁. DEPTH: 7 — elegant unification of Vieta jumping, continued fractions, and Fibonacci sequences; deep but not revolutionary. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com