Vieta Jumping as Affine Weyl Group Orbits on Quadratic Forms — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Vieta jumping is the elementary face of an affine Weyl group orbit structure on integer solutions of quadratic forms; the arXiv paper formalizes the infinite reduced word geometry underlying this. MATH: Vieta jumping on x2 + y2 = kxy + c (or ax2 + bxy + cy2 = n) corresponds to the action of the simple reflection si on the root lattice: (x,y) ↦ (kx − y, x) for k = (x2+y2−c)/xy. This is precisely the Weyl group W(Ã1) (or higher Ãn) acting on the integer points of the quadratic form's level set. The affine Weyl group Ãn has Coxeter generators s0, s1, …, sn with si2=1, (si si+1)3=1, and the orbit of a solution under these reflections is the full integer solution set. The paper's "minimal infinite reduced words" encode the infinite descent/ascent chains of Vieta jumps — each jump is a simple reflection, and the sequence of jumps is a reduced word in the affine Weyl
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 as Affine Weyl Group Orbits on Quadratic Forms — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409733},
url = {https://zenodo.org/records/23409733},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Vieta Jumping as Affine Weyl Group Orbits on Quadratic Forms — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409733
UR - https://zenodo.org/records/23409733
AB - FINDING: Vieta jumping is the elementary face of an affine Weyl group orbit structure on integer solutions of quadratic forms; the arXiv paper formalizes the infinite reduced word geometry underlying this. MATH: Vieta jumping on x^2 + y^2 = kxy + c (or ax^2 + bxy + cy^2 = n) corresponds to the action of the simple reflection s_i on the root lattice: (x,y) ↦ (kx − y, x) for k = (x^2+y^2−c)/xy. This is precisely the Weyl group W(Ã_1) (or higher Ã_n) acting on the integer points of the quadratic form's level set. The affine Weyl group Ã_n has Coxeter generators s_0, s_1, …, s_n with s_i^2=1, (s_i s_i+1)^3=1, and the orbit of a solution under these reflections is the full integer solution set. The paper's "minimal infinite reduced words" encode the infinite descent/ascent chains of Vieta jumps — each jump is a simple reflection, and the sequence of jumps is a reduced word in the affine Weyl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -