不定方程是数论的一个重要分支,指的是未知数多于方程个数的一类方程.它的研究成果广泛应用于数学的各个分支中,在解决生活实际问题时也常常发挥作用.本文主要利用初等代数的方法,研究了线性不定方程解存在的充要条件,解的结构,以及正整数解的数量,并对一般线性不定方程解的组数进行了最大估计. 1.研究了二元与三元线性不定方程的解存在的充要条件,并给出了二元与三元线性不定方程的通解公式,还介绍了如何用辗转相除法求出一组特解. 2.将研究二元与三元线性不定方程的方法推广到一般情形,证明了n元线性不定方程解存在的充要条件,并通过计算给出了通解公式. 3.讨论了二元与三元线性不定方程解的数量,通过引入辅助参数的方法给出了求系数含1的三元线性不定方程解数量的公式. 4.介绍了生成函数法,并将引入辅助参数法推广到一般情形,给出了求系数含1的n元线性不定方程解数量的公式,还对一般线性不定方程解的组数进行了最大估计.

Indeterminate equation is an important branch of number theory. It refers to a class of equations with unknown numbers more than the number of equations. Its research results are widely used in various branches of mathematics, and often play an important role in solving practical problems in life. In this paper, the elementary algebra method is used to study the necessary and sufficient conditions for the existence of solutions of linear indeterminate equations, and the number of positive integer solutions. The maximum number of solutions for general linear indeterminate equations is estimated. Firstly, we discuss the necessary and sufficient conditions for the existence of solutions of binary and ternary linear indeterminate equations. The general solution formulas for binary and ternary linear indeterminate equations are also given. In addition, we introduce how to find a set of special solutions by division algorithm. Secondly, we generalize the method of studying binary and ternary linear indeterminate equations to the general case, and prove the necessary and sufficient conditions for the existence of general linear indeterminate equations, and give the general solution formula through calculation. Thirdly, we discuss the number of solutions of binary and ternary linear indeterminate equations. By introducing the auxiliary parameters, we give the formula for the number of solutions of ternary linear indeterminate equations with coefficients of 1. Finally, we introduce the generator function method, and introduce the auxiliary parameter method to the general case. The formula for finding the number of solutions of general linear indeterminate equations with coefficients is given, and the maximum number of groups of solutions for general linear indeterminate equations is also estimated.