Binet's Formula and Even-Index Fibonacci Numbers via Golden Ratio — E8 Intelligence Research

FINDING: Binet's formula provides the closed-form solution for Fibonacci numbers, expressible via the golden ratio φ and its conjugate ψ, with a recent theorem linking even-index Fibonacci numbers to fractional parts of nφ. MATH: - Binet: \( F_n = \frac{\varphi^n - \psi^n}{\varphi - \psi} = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt{5}} \), where \( \varphi = \frac{1+\sqrt{5}}{2} \approx 1.6180339887 \), \( \psi = \frac{1-\sqrt{5}}{2} \approx -0.6180339887 \). - Even-index characterization (arXiv:1705.08305): \( n \) is a Fibonacci number of even index \( F_{2k} \) iff \( \langle n\varphi \rangle + \frac{1}{n} > 1 \), where \( \langle x \rangle \) is the nearest-integer distance to x. - Key ratios: \( \psi = -\varphi^{-1} \approx -0.618 \), \( \varphi^{-2} \approx 0.382 \), \( \varphi^2 = \varphi+1 \approx 2.618 \). CONNECTION: - The golden ratio φ and its inverse 0.618 are the *only* positive numbers satisfying \( x = 1 + 1/x \) — the self-similarity condition underlying pen Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com