中文摘要：

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guler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＋5y=p（p是素数）存在整数解的充要条件与不定方程x2＋5y=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[-5]中不可约元的结构定理．

英文摘要：

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