Differences between perfect powers: The Lebesgue-Nagell equation
Differences between perfect powers: The Lebesgue-Nagell equation
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完美幂之间的差异：勒贝格-纳格尔方程
DOI：
10.1090/tran/8734
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发表时间：
2022
影响因子：
1.3
通讯作者：
Bennett M
中科院分区：
文献类型：
--
作者：
Bennett M
我们开发了各种新的技术来处理丢番图方程的形状x2+D=yn，基于边界的线性形式的p-adic和复杂的代数，模块化的伽罗瓦表示附加到Frey-Hellegouarch椭圆曲线，和机械丢番图近似。我们使用这些来明确地确定所有互质整数x和y的集合，并且n≥3，具有yn>x2和x2−yn没有超过11的素因子的性质。
We develop a variety of new techniques to treat Diophantine equations of the shape x2+D=yn, 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≥3, with the property that yn>x2 and x2−yn has no prime divisor exceeding 11.