一类不定方程解存在的充要条件及其应用

The Sufficient and Necessary Conditions of Existence for the Integer Solution to a Type of Indeterminate Equation Application

探究了不定方程x2+5y2=n(n∈Z)存在整数解的充分必要条件.运用Euler判别法与Gauss二次互反律等数论的基础知识,先从n为素数p的情况着手讨论,再拓展到n为一般正整数的情况,给出了2个主要结论:不定方程x2+5y2=p(p是素数)存在整数解的充要条件与不定方程x2+5y2=n(n∈Z)存在整数解的充要条件,并利用这2个结果证明了整环Z[槡-5]中不可约元的结构定理.

We prove the sufficient and necessary conditions of existence for the integer solution to a type of indeterminate equations in the form of x2 ＋ 5y2 = n（n∈ N）. Applying some basic knowledge of number theory,such as Euler discriminant analysis and the law of quadratic reciprocity, we first discuss the problem in the case that n is a prime p,and then generalize the prime p to a positive integer n for further discussion. Thus we get the two main results of this paper：which tells when indeterminate equations in the form of x2 ＋5y2 = p（p is a prime） have the integer solution and which gives the sufficient and necessary conditions of existence for the in- teger solution to a types of indeterminate equations in the form of x2 ＋ 5y2 = n（n ∈ N）. In the end,we give their application to the structure of irreducible elements in domain ring Z[ -5]. Key