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

**FINDING:** Vieta jumping is a number theory technique that solves certain Diophantine equations by exploiting symmetry and infinite descent, often producing integer sequences linked to Fibonacci-like recurrences and root lattices. **MATH:** - Core equation: \((a+1)/b + (b+1)/a = k\) (integer \(k\)). - Vieta jumping: given a solution \((a,b)\), treat \(a\) as root of quadratic \(x^2 - (kb-1)x + (b^2+1)=0\); the other root \(a' = kb-1 - a\) yields a smaller positive integer solution, enabling descent. - Recurrence: \(a_{n+1} = k a_n - a_{n-1}\) (Fibonacci-type). - For \(k=3\), solutions correspond to consecutive Fibonacci numbers: \((1,1), (1,2), (2,5), (5,13), \dots\) with ratio \(\phi^2 \approx 2.618\). - Lie algebra \(\mathfrak{sl}(2,\mathbb{Z})\): integer matrices with trace zero; its root system is \(A_1\), with Weyl group \(S_2\) and Coxeter number 2. Vieta jumping's symmetry mirrors the Weyl reflection \(a \mapsto a'\). **CONNECTION:** - **Golden ratio \(\phi \appr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com