The sum of two cubes problem -- an approach that's classroom friendly
| Autor: | Monsky, Paul |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 18 pages |
| Databáze: | arXiv |
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