Algebraic Units and Quintic Symmetry in Rogers-Ramanujan Continued Fraction Values — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Rogers-Ramanujan continued fraction (RRCF) special values are algebraic units in Q(√5), linked to quintic Galois group A5 via modular equations and elliptic integrals. | MATH: RRCF: R(q) = q^{1/5}/(1 + q/(1 + q²/(1 + q³/(...)))). For q = e^{-2π}, R(q) = √( (5−√5)/2 ) − (√5−1)/2 ≈ 0.09017... (algebraic unit in Q(√5)). Key identity: 1/R(q) − 1 − R(q) = η(q^{1/5})/η(q^{5}) (eta-quotient). Special values at q = e^{−2π√n} yield units in real quadratic fields, often Q(√5). Quintic connection: RRCF solves the Bring–Jerrard quintic x⁵ + x = a via modular lambda and elliptic integrals; Galois group of the resolvent is A5 (icosahedral). | CONNECTION: The golden ratio φ = (1+√5)/2 ≈ 1.618 appears explicitly: RRCF at q = e^{−2π} involves √5 and φ. The algebraic unit structure in Q(√5) is the ring Z[φ] — the same lattice used in Penrose tilings (5-fold crystallographic symmetry, forbidden in periodic crystals but allowed in quasicrystals). The A5 Galois group is the rotational symmetry gro
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026algebraic,
title = {{Algebraic Units and Quintic Symmetry in Rogers-Ramanujan Continued Fraction Values — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23446676},
url = {https://zenodo.org/records/23446676},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebraic Units and Quintic Symmetry in Rogers-Ramanujan Continued Fraction Values — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23446676
UR - https://zenodo.org/records/23446676
AB - FINDING: Rogers-Ramanujan continued fraction (RRCF) special values are algebraic units in Q(√5), linked to quintic Galois group A5 via modular equations and elliptic integrals. | MATH: RRCF: R(q) = q^{1/5}/(1 + q/(1 + q²/(1 + q³/(...)))). For q = e^{-2π}, R(q) = √( (5−√5)/2 ) − (√5−1)/2 ≈ 0.09017... (algebraic unit in Q(√5)). Key identity: 1/R(q) − 1 − R(q) = η(q^{1/5})/η(q^{5}) (eta-quotient). Special values at q = e^{−2π√n} yield units in real quadratic fields, often Q(√5). Quintic connection: RRCF solves the Bring–Jerrard quintic x⁵ + x = a via modular lambda and elliptic integrals; Galois group of the resolvent is A5 (icosahedral). | CONNECTION: The golden ratio φ = (1+√5)/2 ≈ 1.618 appears explicitly: RRCF at q = e^{−2π} involves √5 and φ. The algebraic unit structure in Q(√5) is the ring Z[φ] — the same lattice used in Penrose tilings (5-fold crystallographic symmetry, forbidden in periodic crystals but allowed in quasicrystals). The A5 Galois group is the rotational symmetry gro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -