多元一次不定方程的整数解

多元1次不定方程的整数解
 

摘要

这篇主要讨论多元1次不定方程有整数解的充要条件、整数解的求法以及有非负整数解的条件，在此基础上进1步探讨当2元1次不定方程ax+by=c可解时，它有负整数解的条件．得到了如下这个结论：设a，b都为正整数，c为负整数，(a , b)=1，那么当c＜－(ab－a－b)时，2元1次不定方程ax+by=c (1)有负整数解，负整数解的个数等于－[c／(ab)]－1或－[c／(ab)]，当c≥ab－a－b时，2元1次不定方程(1)无负整数解．
关键词：2元1次不定方程；多元1次不定方程；整除；最大公因数；整数解．

The Integer Solutions of
Multivariate Indeteminate Equation of the First Degree
 

ABSTRACT

This paper mainly studies multivariate indeteminate equation of the first degree have necessary and sufficent condition of integer solutions and extraction of integer solutions as well as the condition of the non-negative integer roots. Regarding this I carry on further study, when discussed about the integer solutions of duality indeteminable equation of the first degree, it has the condition of the negative integer roots. Therefore, I get the following conclusion: Supposes a, b both are the positive integers, c is a negative integer, and (a , b) =1, then in terms of c＜－(ab－a－b), duality indeteminable equation of the first degree ax+by=c (1) has the negative integer roots, and the number of negative integer roots is equal to －[c／(ab)]－1 or －[c／(ab)].When c≥ab－a－b, duality indeteminable equation of the first degree(1) has not the negative integer roots.
Key words: Duality indeteminable equation of the first degree;  Multivariate indeteminate equation of the first degree;  Divisible;  Greatest common divisor;  Integer solution.