Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Diophantine equations with sum of cubes and cube of sum   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 ratio of cubes 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 certain restrictions on the integer $m$. The equations can be
represented by elliptic curves unless $a = 9$ or 1. If $a$ is an integer and
two variables are equal and nonzero, there exist nontrivial solutions only for
$a=4$ or 9; there are no solutions for $a = 4$ when $xyz \neq 0$. Without
imposing constraints on the variables, we find the general solution for $a =
9$, 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.

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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: cs
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas