Abstract
This paper contains an explanation of Ramanujan-type formulas with cubic radicals of cubic irrationalities in the situation when these radicals are contained in a pure cubic extension. We give a complete description of formulas of such type, answering the Zippel’s question. It turns out that Ramanujan-type formulas are in some sense unique in this situation. In particular, there must be no more than three summands in the right-hand side and the norm of the irrationality in question must be a cube. In this situation we associate cubic irrationalities with a cyclic cubic polynomial, which is reducible if and only if one can simplify the corresponding cubic radical. This correspondence is inverse to the so-called Ramanujan correspondence defined in the preceding papers, where one associates a pure cubic extension to some cyclic polynomial.
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To Alexander Generalov with gratitude and appreciation
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Antipov, M.A., Pimenov, K.I. Ramanujan Denesting Formulas for Cubic Radicals. Vestnik St.Petersb. Univ.Math. 53, 115–121 (2020). https://doi.org/10.1134/S1063454120020028
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DOI: https://doi.org/10.1134/S1063454120020028