摘 要
丢番图方程也称为不定方程，是指变量个数多于方程个数，且取整数值的方程(或者方程组)。它是数论的一个分支，有着广泛的研究价值，与代数几何、组合数学、代数编码等学科有着紧密的联系。它的研究成果不仅仅对数学的各个分支起着推动作用，而且对非数学学科诸如计算机科学、电子学、信号的数字处理等有重大的实际意义。求解丢番图方程的方法较多，既有初等方法也有高等方法。但是已经证明：不可能存在一个通用的算法能够判定所有的丢番图方程是否有解，这就给研究人员带来了很大的难度。
本文对不定方程的整数解问题做了相应的研究，得出了相应的结论并对此作出证明。第一章介绍了不定方程的一系列发展概况、实际应用；第二章主要讲述了不定方程的相关概念和一些不定方程的在相关性质方面的例子；第三章对不定方程的整数解做了相应的研究，并作出了相应的证明。
关键词 不定方程；同余式；整数解；递归序列；平方剩余
Solution of indeterminate equation
Abstract
The Diophantine equation is also called the indeterminate equation. It means that the number of variables is more than the number of equations, also, the equation of the whole thing (or the system of equations) It is a branch of mathematical theory, with extensive research value, and closely related to algebra geometry, combinatorial mathematics, and algebraic coding. This research not only plays an important role in every branch of mathematics, but also have great practical significance for the mathematical disciplines such as computer science, electronics, digital signal processing etc. There are many methods in solving the Diophantine equations, both elementary methods and higher methods. But it has been prove that there is impossible a general algorithm can determine all the Diophantine equations whether there is a solution or not. It brings many difficulties to the researchers.
This paper studies the integer solution of equation The corresponding conclusions are drawn and the proof is made. The first chapter introduces a series of developments and practical applications of the indefinite equation. The second chapter focuses on the concepts of the indefinite equation and the relevant properties of some of the indeterminate equations. In the third chapter, the integral solution of the indeterminate equation is studied, and the corresponding proof is made.
Keywords Diophantine equation; Congruence type; Integer solutions; Recursion sequence; Square residual
1968年，英国数学家A.Baker给出了方程 解的一个可计算的上界，并成功的将Gelfond和Schncider有关Hilbert第七个问题的结果推广到一般的情况，从而确定了一大类丢番图方程的整数解的绝对值的上界。Baker的工作不但推动了超越数论的发展，而且给数论中包括丢番图方程在内的许多领域带来了突破性进展。
1969年，“不定方程之王”L.Jmordell详细地总结了当时的成果，写成了著名的《丢番图方程》(Dipohantine Equations，Ademie Press)。
1973年，P.Deligne以清晰简洁地证明解决了有限域上丢番图方程解的个数的猜想，也就是举世闻名的的A.Weil猜想。
1983年，德国数学家G .Faltings用代数几何学的方法证明了在数论中莫德尔猜想，毫无疑问，这是20世纪数论中最杰出的工作之一，即在有理数域里亏格>1的代数曲线上仅有有限个有理点。由此可以推出Fcrmat方程+在时最多有有限组整数解。D.R.Heath Brown证明了对“几乎所有”的正整数>2，方程均没有正整数解。
