This article is cited in 1 paper

ON THE ANNIVERSARY OF A. I. GENERALOV

Ramanujan denesting formulae for cubic radicals

M. A. Antipov^ab, K. I. Pimenov<sup>b

a</sup> National Research University Higher School of Economics, 16, ul. Soyuza Pechatnikov, St. Petersburg, 190121, Russian Federation
^b St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

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 with cubic irrationalities a cyclic cubic polinomial, 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 polinomial.

Keywords: Ramanujan formulas, simplification of radicals, Ramanujan correspondence.

UDC: 512.62

MSC: 12F02

Received: 18.11.2019
Revised: 12.12.2019
Accepted: 12.12.2019

DOI: 10.21638/11701/spbu01.2020.201

 English version: Vestnik St. Petersburg University, Mathematics, 2020, 7:2, 115–121