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      The sum of two cubes problem -- an approach that's classroom friendly

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          Abstract

          In this note I give simple proofs of classical results of Euler, Legendre and Sylvester showing that for certain integers M there are no (or only a few) solutions of \(x^3 + y^3 = M\), with \(x\) and \(y\) in \(\mathbb{Q}\). The proofs all use a single argument -- infinite 3-descent in the ring \(\mathcal{O} = \mathbb{Z}[\omega]\) of Eisenstein integers. (Everything needed about \(\mathcal{O}\) is developed from scratch.) The reader only needs the briefest acquaintance with complex numbers, fields and congruence modulo an element of a commutative ring. In particular I never say anything about ideals or elliptic curves (though I do mention cubic reciprocity in passing), and a clever high-school student might well enjoy the note. A few new results with \(M\) in \(\mathcal{O}\) and \(x\) and \(y\) in \(\mathbb{Q}[\omega]\) are also derived.

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          Author and article information

          Journal
          31 August 2023
          Article
          2309.00162
          9b85bd38-fb79-4aa0-a484-f57501aaadb3

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          18 pages
          math.HO math.NT

          History & Philosophy,Number theory
          History & Philosophy, Number theory

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