Rogers-Ramanujan Continued Fraction Solves Quintics, Yields Golden Ratio — E8 Intelligence Research

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The Rogers-Ramanujan continued fraction is a modular form that can solve the general quintic equation, with its evaluation at specific arguments yielding the golden ratio and related algebraic numbers. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22138308

August 28, 2026Open Access

Rogers-Ramanujan Continued Fraction Solves Quintics, Yields Golden Ratio — E8 Intelligence Research

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Key Points

To evaluate how the Rogers-Ramanujan continued fraction can analytically solve general quintic equations and establish algebraic links to the golden ratio.
Analyzed the analytic structure and q-series infinite product identities of the Rogers-Ramanujan continued fraction R(q).
Mapped Bring-Jerrard quintic equation coefficients to modular parameters using elliptic integrals and modular lambda functions.
Demonstrated that the general quintic equation x^5 + ax + b = 0 can be solved analytically as an algebraic function of R(q).
Showed that degree-5 modular equations satisfied by R(q) directly relate special evaluations such as R(e^(-2pi)) to algebraic forms of the golden ratio phi.

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Implication

Theoretical analysis demonstrates algebraic solutions for general quintic equations via continued fractions, highlighting direct mathematical connections to the golden ratio.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a93f0ce6c1a8fb52e79d3e0 https://doi.org/10.5281/zenodo.22138308

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Rogers-Ramanujan Continued Fraction Solves Quintics, Yields Golden Ratio — E8 Intelligence Research

Key Points

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Implication

Discussion

Cite This Study

Rogers-Ramanujan Continued Fraction Solves Quintics, Yields Golden Ratio — E8 Intelligence Research

Key Points

No abstract yet

Implication

Discussion

Cite This Study

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