摘要
运用同余理论、因式分解、数学归纳法和Legendre符号等基础知识,得到了不定方程x^(2)+7y^(2)=n(n∈N^(*))有互素的正整数解的充分必要条件,证明了方程有解时,恰有2m-1个解,这里m是n的不同素因子的个数,并给出了解的形式.最后利用结论证明了整环Z[√-1]中不可约元的充要条件.
In this paper,by using the basic knowledge of congruence theory,factorization,mathematical induction and Legendre symbol,we obtain the sufficient and necessary conditions for positive integer solutions of the reciprocal prime of the indefinite equation x^(2)+7 y^(2)=n(n∈N^(*)).It is proved that when the equation has a solution,there are exactly 2m-1 solutions,where m is the number of different prime factors of n,and the form of the solution is given.Finally,the sufficient and necessary conditions of irreducible elements in domain ring Z[√-1] are proved by using the conclusion.
作者
陈云凤
罗家贵
CHEN Yun-feng;LUO Jia-gui(School of Mathematic and Information,China West Normal University,Nanchong 637009,China)
出处
《数学的实践与认识》
2021年第17期287-297,共11页
Mathematics in Practice and Theory
基金
四川省教育厅重点项目(16ZA0173)
国家自然科学基金(11871058)。
关键词
不定方程
整数解
数学归纳法
整环
不可约元
indefinite equation
integer solutions
mathematical induction
domain ring
irreducible element
