Golden Ratio BBP Formulas for Pi and Arctangents — E8 Intelligence Research

Abstract

FINDING: The arXiv paper (1603.06307v1) is the sole substantive mathematical result — deriving BBP-type digit-extraction formulas for arctangents of odd powers of the golden ratio, plus a golden-ratio-base BBP formula for π. The YouTube links are pedagogical or audio-sensory, not novel mathematics. MATH: - BBP formula (Bailey–Borwein–Plouffe, 1997): π = Σ_{k=0}^∞ (1/16^k) [4/(8k+1) − 2/(8k+2) − 1/(8k+4) − 1/(8k+6)] — permits base-16 digit extraction without prior digits. - The paper generalizes this to arctangent identities: arctan(φ^(−m)) for odd m, where φ = (1+√5)/2 ≈ 1.6180339887. Example identity (from abstract context): π/4 = arctan(1/φ) + arctan(1/φ³) + arctan(1/φ⁵) + … (Machin-like, telescoping via Fibonacci/Lucas identities). - Golden-ratio-base BBP: replace base 16 with base φ² (or φ), yielding digit extraction in a non-integer base. - Fibonacci/Lucas recurrence coefficients: F_n = (φⁿ − ψⁿ)/√5, ψ = (1−√5)/2 = −1/φ; L_n = φⁿ + ψⁿ. Arctangent i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com