Complete Solution of the Diophantine EquationX2+1=dY4and a Related Family of Quartic Thue Equations☆
Complete Solution of the Diophantine EquationX2+1=dY4and a Related Family of Quartic Thue Equations☆
复制标题
丢番图方程X2+1=dY4及相关四次Thue方程族的完全解☆
DOI：
10.1006/jnth.1997.2018
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
P. Voutier
中科院分区：
文献类型：
--
作者：
C. Hua;P. Voutier
本文利用Thue和Siegel的方法，基于代数函数的显式Pade逼近，完整地求解了一族四次Thue方程。根据这个结果，我们还可以解标题中的丢番图方程。我们证明了当⩾为3时，该方程至多有一个正整数解。而且，当存在这样的解时，它的形式是(u，v)，其中(u，v)是X2+1=dY2的基本解。
In this paper, we use the method of Thue and Siegel, based on explicit Pade approximations to algebraic functions, to completely solve a family of quartic Thue equations. From this result, we can also solve the diophantine equation in the title. We prove that this equation has at most one solution in positive integers whend⩾3. Moreover, when such a solution exists, it is of the form(u, v)where (u, v) is the fundamental solution ofX2+1=dY2.