厦门大学学报（自然科学版）2013，Vol.52Issue(4)：441-446,6.DOI:10.6043/j.issn.0438-0479.2013.04.002
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
摘要
Abstract
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 integer 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].关键词
整环/不定方程/Legendre符号Key words
domain/indeterminate equation/Legendre symbol分类
数理科学引用本文复制引用
林桂娟,辛林..一类不定方程解存在的充要条件及其应用[J].厦门大学学报（自然科学版）,2013,52(4):441-446,6.基金项目
国家自然科学基金项目(11071040) （11071040）
福建省自然科学基金项目(2011J01004) （2011J01004）
