完全幂之间的差异：Lebesgue-Nagell方程
Differences between perfect powers : the Lebesgue-Nagell Equation
摘要：
我们开发了各种新的技术来处理丢番图方程的形状$x^2+D =y^n$，基于边界的线性形式在$p$进和复杂的矩阵，模块化的伽罗瓦表示附加到Frey-Hellegouarch椭圆曲线，和机械从丢番图逼近。我们使用这些显式地确定所有互质整数$x$和$y$，和$n \geq 3$的集合，具有$y^n > x^2$和$x^2-y^n$没有超过$11$的素因子的性质。
英文摘要：
We develop a variety of new techniques to treat Diophantine equations of the shape $x^2+D =y^n$, based upon bounds for linear forms in $p$-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers $x$ and $y$, and $n \geq 3$, with the property that $y^n > x^2$ and $x^2-y^n$ has no prime divisor exceeding $11$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
