Perfect Cuboids and Elliptic Curves: A Degree-8 Polynomial Link — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The perfect cuboid problem is linked to irreducibility of a degree-8 "cuboid polynomial" via a rank-zero elliptic curve, while quartic models of elliptic curves govern rational Diophantine D(q)-quintuples. | MATH: Cuboid polynomial Pa,u(t) (even, monic, degree 8) from Sharipov's first-cuboid specialization; irreducibility in ℤ[t] proven for nonzero integers a,u with u2 ≠ a2 by reducing to a rank-zero elliptic curve. Quartic model: y2 = x4 + a x3 + b x2 + c x + d (or y2 = a x4 + b x3 + c x2 + d x + e) — divisibility-by-2 maps on such quartics relate to Diophantine D(q)-quintuples (sets {a1,...,a5} where ai aj + q is square). | CONNECTION: Rank-zero elliptic curves have finite Mordell–Weil group — their torsion structure connects to root systems (e.g., E8 lattice via j=0 curves, or A4 from quintuple symmetries). The degree-8 polynomial's even symmetry hints at D4 (hyperoctahedral) symmetry, a c
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026perfect,
title = {{Perfect Cuboids and Elliptic Curves: A Degree-8 Polynomial Link — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587298},
url = {https://zenodo.org/records/23587298},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Perfect Cuboids and Elliptic Curves: A Degree-8 Polynomial Link — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587298
UR - https://zenodo.org/records/23587298
AB - FINDING: The perfect cuboid problem is linked to irreducibility of a degree-8 "cuboid polynomial" via a rank-zero elliptic curve, while quartic models of elliptic curves govern rational Diophantine D(q)-quintuples. | MATH: Cuboid polynomial P_a,u(t) (even, monic, degree 8) from Sharipov's first-cuboid specialization; irreducibility in ℤ[t] proven for nonzero integers a,u with u^2 ≠ a^2 by reducing to a rank-zero elliptic curve. Quartic model: y^2 = x^4 + a x^3 + b x^2 + c x + d (or y^2 = a x^4 + b x^3 + c x^2 + d x + e) — divisibility-by-2 maps on such quartics relate to Diophantine D(q)-quintuples (sets {a_1,...,a_5} where a_i a_j + q is square). | CONNECTION: Rank-zero elliptic curves have finite Mordell–Weil group — their torsion structure connects to root systems (e.g., E_8 lattice via j=0 curves, or A_4 from quintuple symmetries). The degree-8 polynomial's even symmetry hints at D_4 (hyperoctahedral) symmetry, a c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -