Theon's Recurrence: Pell Numbers, √2 Approximations, and Golden Ratio Links — E8 Intelligence Research

Abstract

FINDING: Theon of Smyrna's side-and-diagonal number recurrence generates successive rational approximations to √2, which are also linked to Pell numbers and the Fibonacci-like growth of the golden ratio. MATH: - Recurrence: \( a_{n+1} = a_n + 2b_n \), \( b_{n+1} = a_n + b_n \) (with \( a_1 = 1, b_1 = 1 \)). - Ratio \( b_n / a_n \) → √2 as \( n \to \infty \). Sequence: 1/1, 3/2, 7/5, 17/12, 41/29, … - Pell numbers: \( P_n \) satisfy \( P_{n+1} = 2P_n + P_{n-1} \); here \( a_n = P_n \), \( b_n = P_{n+1} \). - Connection to Fibonacci: Both sequences share the same recurrence form \( x_{n+1} = 2x_n + x_{n-1} \) (Pell) vs \( F_{n+1} = F_n + F_{n-1} \) (Fibonacci). The golden ratio φ = (1+√5)/2 ≈ 1.618; √2 ≈ 1.414. - Key constant: √2 = 1.41421356…; φ = 1.61803398…; ratio φ/√2 ≈ 1.144. CONNECTION: - The recurrence mirrors the geometric construction of a square's diagonal (side = a, diagonal = b). The ratio b/a → √2, the fundamental irrational in square geometry. - The golden r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com