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: \( F_n = φ^n - ψ^n/φ - ψ = {φ^n - (-φ)⁻ⁿ}{√5} \), where \( φ = {1+√5}{2} ≈ 1.6180339887 \), \( ψ = {1-√5}{2} ≈ -0.6180339887 \). - Even-index characterization (arXiv:1705.08305): \( n \) is a Fibonacci number of even index \( F₂ₖ \) iff \( nφ + 1/n > 1 \), where \( x \) is the nearest-integer distance to x. - Key ratios: \( ψ = -φ⁻¹ ≈ -0.618 \), \( φ⁻² ≈ 0.382 \), \( φ^2 = φ+1 ≈ 2.618 \). 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