Nested fifth root radical identities from elliptic curves

Abstract

Ramanujan discovered the following elegant identity involving cube roots:$\sqrt{m\sqrt[3]{4(m-2n)}+n\sqrt[3]{4m+n}}=\pm \frac{1}{3}(\sqrt[3]{{(4m+n)}^2}+\sqrt[3]{4(m-2n)(4m+n)}-\sqrt[3]{2{(m-2n)}^2}).$

The goal of this paper is to derive nested fifth root radical identities in the form of$\sqrt{P_1\sqrt[5]{Q_1}+P_2\sqrt[5]{Q_2}}=p_1\sqrt[5]{q_1}+p_2\sqrt[5]{q_2}+p_3\sqrt[5]{q_3}+p_4\sqrt[5]{q_4}+p_5\sqrt[5]{q_5}.$ We show that these identities can be derived from a specific family of elliptic curves. Furthermore, we include the SageMath code for computing these expressions.