论文标题
$ \ mathbb {q} $ - 曲线和lebesgue-nagell方程
$\mathbb{Q}$-curves and the Lebesgue-Nagell equation
论文作者
论文摘要
在本文中，我们考虑方程\ [x^2 -q^{2k+1} = y^n，\ qquad q \ nmid x，\ quad 2 \ quad 2 \ mid y，\]对于整数$ x，q，q，k，y $和$ n $，带有$ k \ geq 0 $ and $ n $ and $ n \ geq 3 $。我们通过在$ q = 41 $和$ q = 97 $的情况下找到所有解决方案来扩展第一和第三名作者的工作。我们通过构建Frey-hellegouarch $ \ Mathbb {q} $ - 在实际二次字段$ k = \ mathbb {q}（\ sqrt {q}）$上定义的曲线，并使用模块化方法与多折线技术。
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.
