Ramanujan's Continued Fractions and Quartic Modular Conjugates — E8 Intelligence Research

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Ramanujan's continued fractions yield quartic polynomial conjugates with modular equation structure, linking to elliptic integrals and algebraic integers. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22230160

September 1, 2026Open Access

Ramanujan's Continued Fractions and Quartic Modular Conjugates — E8 Intelligence Research

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Key Points

To examine the algebraic properties and modular equation frameworks of Ramanujan's continued fractions parameterized by elliptic integrals.
Evaluated degree-5 modular equations governing the Rogers-Ramanujan continued fraction R(q) and related generalized quantities.
Parameterized continued fractions using generalized elliptic integrals where q is expressed as an exponential function across the upper half-plane.
Showed that distinct continued fractions, including R(q) and R(q²), form algebraic conjugates that satisfy the identical quartic equation ax⁴ + bx³ + cx² + dx + e = 0 with integer coefficients.
Identified that roots of the quartic modular conjugates converge to the golden ratio φ (~1.618) and 1/φ (~0.618) as q approaches 1, corresponding to degree-5 icosahedral symmetry.

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Implication

Theoretical analysis reveals algebraic quartic polynomial conjugates within Ramanujan continued fractions, highlighting underlying icosahedral symmetry and modular equation structures.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9935d8636c6408cfa7e68f https://doi.org/10.5281/zenodo.22230160

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Ramanujan's Continued Fractions and Quartic Modular Conjugates — E8 Intelligence Research

Key Points

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Discussion

Cite This Study

Ramanujan's Continued Fractions and Quartic Modular Conjugates — E8 Intelligence Research

Key Points

No abstract yet

Implication

Discussion

Cite This Study

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