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| Formato: | Preprint |
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2020
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| Acceso en línea: | https://arxiv.org/abs/2012.04139 |
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| author | Dobrescu, Bogdan A. Fox, Patrick J. |
| author_facet | Dobrescu, Bogdan A. Fox, Patrick J. |
| contents | We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any cube or certain rational fractions, that have nontrivial solutions. There are also infinite families of equations that do not have any nontrivial solution, including those where $1/a = 1- 24/m$ with restrictions on the integer $m$. The equations can be represented by elliptic curves unless $a = 9$ or 1, and any elliptic curve of nonzero $j$-invariant and torsion group $\mathbb{Z}/3k\mathbb{Z}$ for $k = 2,3,4$, or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z} $ corresponds to a particular $a$. We prove that for any $a$ the number of nontrivial solutions is at most 3 or is infinite, and for integer $a$ it is either 0 or $\infty$. For $a = 9$, we find the general solution, which depends on two integer parameters. These cubic equations are important in particle physics, because they determine the fermion charges under the $U(1)$ gauge group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_04139 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Diophantine equations with sum of cubes and cube of sum Dobrescu, Bogdan A. Fox, Patrick J. Number Theory High Energy Physics - Phenomenology High Energy Physics - Theory 11D25 (11G05, 11D45, 11D85) We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any cube or certain rational fractions, that have nontrivial solutions. There are also infinite families of equations that do not have any nontrivial solution, including those where $1/a = 1- 24/m$ with restrictions on the integer $m$. The equations can be represented by elliptic curves unless $a = 9$ or 1, and any elliptic curve of nonzero $j$-invariant and torsion group $\mathbb{Z}/3k\mathbb{Z}$ for $k = 2,3,4$, or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z} $ corresponds to a particular $a$. We prove that for any $a$ the number of nontrivial solutions is at most 3 or is infinite, and for integer $a$ it is either 0 or $\infty$. For $a = 9$, we find the general solution, which depends on two integer parameters. These cubic equations are important in particle physics, because they determine the fermion charges under the $U(1)$ gauge group. |
| title | Diophantine equations with sum of cubes and cube of sum |
| topic | Number Theory High Energy Physics - Phenomenology High Energy Physics - Theory 11D25 (11G05, 11D45, 11D85) |
| url | https://arxiv.org/abs/2012.04139 |