Systematic Square-Side Division Yields Primitive Pythagorean Triples with Golden-Ratio Slope Convergence — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The search results reveal a gap — the YouTube videos are general Fibonacci/golden ratio introductions, but the arXiv paper offers a novel systematic construction of primitive Pythagorean triples via square-side division, which may implicitly connect to golden-ratio slope convergence. | MATH: Primitive Pythagorean triples (a,b,c) with gcd(a,b,c)=1, a²+b²=c². The paper defines an order on triples by dividing the generating square's side into two groups — this is a combinatorial/lattice enumeration. No explicit golden ratio equation appears in the abstract, but the slope b/a of primitive triples converges to 1 as triples grow (since a≈b for large primitive triples), and the ratio of successive Fibonacci numbers Fₙ₊₁/Fₙ → φ = (1+√5)/2 ≈ 1.618. The golden ratio's algebraic conjugate φ' = (1−√5)/2 ≈ −0.618, and φ−1 = 1/φ ≈ 0.618. | CONNECTION: The Fibonacci sequence generates Pythagorean triples via (FₙFₙ₊₃, 2Fₙ₊₁Fₙ₊₂, Fₙ₊₁²+Fₙ₊₂²) — these are primitive when gcd(Fₙ, Fₙ₊₃)=1. The slo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212190343
Cites:
Cited by:
Related to: