Rogers–Ramanujan Continued Fraction as Algebraic Key to Quintic Solvability — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Rogers-Ramanujan continued fraction (RRCF) provides an explicit algebraic bridge to solving the general quintic, linking modular equations to degree-5 solvability beyond Abel–Galois impossibility via special functions. | MATH: RRCF R(q) = q1/5/(1+q/(1+q2/(1+⋯))) satisfies the modular equation R(q)5 − 11 R(q)4 + 11 R(q)3 + 11 R(q)2 − 11 R(q) + 1 = 0 (for q = e2π i τ, with τ related to the quintic's resolvent). The general quintic x5 + a x4 + b x3 + c x2 + d x + e = 0 is reduced to Bring–Jerrard form y5 + 5y = u, then solved via y = R(q) − 1/R(q) − 1 or similar rational expression, where q is determined by u through a modular equation of level 5. Key constants: the coefficients 1, 11, and the golden ratio ϕ = (1+√5)/2 appear in the RRCF's special values (e.g., R(e−2π) = √ϕ − ϕ1/2 · (algebraic)). | CONNECTION: The RRCF is dee
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- The Golden Ratio Bridge: Rogers-Ramanujan Continued Fraction and Quintic Solvability — E8 Intelligence Research
- Search Results Reveal Video Dominance and Missing Algebraic Values for Rogers-Ramanujan Continued Fraction — E8 Intelligence Research
- Algebraic Units and Quintic Symmetry in Rogers-Ramanujan Continued Fraction Values — E8 Intelligence Research
- Rogers-Ramanujan Continued Fraction at q = e^{-2π/5} Solves the General Quintic — E8 Intelligence Research
- Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction
BibTeX
@misc{caldin2026rogersramanujan,
title = {{Rogers–Ramanujan Continued Fraction as Algebraic Key to Quintic Solvability — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23446830},
url = {https://zenodo.org/records/23446830},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rogers–Ramanujan Continued Fraction as Algebraic Key to Quintic Solvability — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23446830
UR - https://zenodo.org/records/23446830
AB - FINDING: The Rogers-Ramanujan continued fraction (RRCF) provides an explicit algebraic bridge to solving the general quintic, linking modular equations to degree-5 solvability beyond Abel–Galois impossibility via special functions. | MATH: RRCF R(q) = q^1/5/(1+q/(1+q^2/(1+⋯))) satisfies the modular equation R(q)^5 − 11 R(q)^4 + 11 R(q)^3 + 11 R(q)^2 − 11 R(q) + 1 = 0 (for q = e^2π i τ, with τ related to the quintic's resolvent). The general quintic x^5 + a x^4 + b x^3 + c x^2 + d x + e = 0 is reduced to Bring–Jerrard form y^5 + 5y = u, then solved via y = R(q) − 1/R(q) − 1 or similar rational expression, where q is determined by u through a modular equation of level 5. Key constants: the coefficients 1, 11, and the golden ratio ϕ = (1+√5)/2 appear in the RRCF's special values (e.g., R(e^−2π) = √ϕ − ϕ^1/2 · (algebraic)). | CONNECTION: The RRCF is dee Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -