Vieta Jumping Reveals Affine Weyl Orbits in Quadratic Diophantine Equations — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Vieta jumping on quadratic Diophantine equations reveals hidden affine Weyl group orbits, with the rational equation (a+1)/b + (b+1)/a = k forcing k ∈ {3,4} and solutions tied to Fibonacci-type recurrences. | MATH: Equation: (a+1)/b + (b+1)/a = k → a² + a + b² + b = kab. Vieta jumping: if (a,b) solves, then (a, ka−b−1) also solves (root reflection). Classification: k=3 or k=4 only. For k=3, solutions follow recurrence a_{n+1} = 3a_n − a_{n−1} − 1 (Fibonacci-like with offset); for k=4, a_{n+1} = 4a_n − a_{n−1} − 1. The quadratic form a² + b² − kab + a + b = 0 has discriminant Δ = k² − 4, so k=3 → Δ=5 (golden ratio base), k=4 → Δ=12 (2√3, hexagonal lattice). | CONNECTION: k=3 gives Δ=5 → √5 appears → ratio (1+√5)/2 = φ ≈ 1.618, and its inverse 0.618. The Vieta jump is a reflection in the affine Weyl group of type A₁ (or Ã₁) — the same group underlying the Fibonacci/Lucas recurrences and the golden ratio's continued fraction [1;1,1,...]. k=4 gives Δ=12 → √12 = 2√3 → ratio √3 ≈ 1.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026vieta,
title = {{Vieta Jumping Reveals Affine Weyl Orbits in Quadratic Diophantine Equations — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23447046},
url = {https://zenodo.org/records/23447046},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Vieta Jumping Reveals Affine Weyl Orbits in Quadratic Diophantine Equations — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23447046
UR - https://zenodo.org/records/23447046
AB - FINDING: Vieta jumping on quadratic Diophantine equations reveals hidden affine Weyl group orbits, with the rational equation (a+1)/b + (b+1)/a = k forcing k ∈ {3,4} and solutions tied to Fibonacci-type recurrences. | MATH: Equation: (a+1)/b + (b+1)/a = k → a² + a + b² + b = kab. Vieta jumping: if (a,b) solves, then (a, ka−b−1) also solves (root reflection). Classification: k=3 or k=4 only. For k=3, solutions follow recurrence a_{n+1} = 3a_n − a_{n−1} − 1 (Fibonacci-like with offset); for k=4, a_{n+1} = 4a_n − a_{n−1} − 1. The quadratic form a² + b² − kab + a + b = 0 has discriminant Δ = k² − 4, so k=3 → Δ=5 (golden ratio base), k=4 → Δ=12 (2√3, hexagonal lattice). | CONNECTION: k=3 gives Δ=5 → √5 appears → ratio (1+√5)/2 = φ ≈ 1.618, and its inverse 0.618. The Vieta jump is a reflection in the affine Weyl group of type A₁ (or Ã₁) — the same group underlying the Fibonacci/Lucas recurrences and the golden ratio's continued fraction [1;1,1,...]. k=4 gives Δ=12 → √12 = 2√3 → ratio √3 ≈ 1. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -