以下为句子列表：
英文: An elementary proof of the Diophantine equation (The equation abbreviated) is given by using recursion sequence method.
中文: 摘要运用递推序列法，给出组合数丢番图方程(方程式略)的一个初等解法。
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英文: By elementary rank transformations of matrix, a method has been obtained to solve linear diophantine equation with some variables. The method overcomes the deficiency of traditional method.
中文: 摘要利用矩阵的初等列变换，给出了求解多元线性不定方程的一种方法，该方法改进了传统方法计算量大、步骤多的缺点。
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英文: Letting Z be integer number ring and A whose normal form is D=VAU=diag(d,d,dq,0,0),this paper obtains an integral solution set to linear Diophantine equation system by means of normal form of matrix A.
中文: 借助Z上矩阵A∈Mm×n(Z)的标准形式D=VAU=diag(d,d,…,dq,0,…,0),得到了整环上不定方程组AX=B的通解公式以及矩阵初等变换法.
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