Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $\mathbb{Q}$-curves and the Lebesgue-Nagell equation   In this paper, we consider the equation \[ x^2 - q^{2k+1} = y^n, \qquad q
\nmid x, \quad 2 \mid y, \] for integers $x,q,k,y$ and $n$, with $k \geq 0$ and
$n \geq 3$. We extend work of the first and third-named authors by finding all
solutions in the cases $q= 41$ and $q = 97$. We do this by constructing a
Frey-Hellegouarch $\mathbb{Q}$-curve defined over the real quadratic field
$K=\mathbb{Q}(\sqrt{q})$, and using the modular method with multi-Frey
techniques.

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The categories of this paper are: atom-ph
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
16
The categories of this paper are: stat
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: eess