Abaffy, Broyden and Spediacto (ABS) introduced a class of the so-called ABS methods to solve systems of linear equations. The ABS approach was specialized to solve linear Diophantine systems by Esmaeili, Mahdavi-Amiri and Spedicato. The method was extended to systems of certain linear inequalities to provide all solutions. Here, we mainly focus on the theoretical research into rank reduction processes for solving linear Diophantine systems and computing integer factorizations. Basic integer ABS algorithm, integer extended ABS (IEABS) algorithm, rank one perturbed problem, the generalized Rosser’s approach (GRA), the extended integer rank reduction process, the integer Wedderburn rank reduction formula and the associated integer biconjugation process are studied. We present an integrated integer rank reduction process, develop various integer factorizations and show their use in solving Diophantine equations.

中文翻译：

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求解线性Diophantine系统和整数分解的降阶过程：综述

Abaffy，Broyden和Spediacto（ABS）引入了一类所谓的ABS方法来求解线性方程组。ABS方法由Esmaeili，Mahdavi-Amiri和Spedicato专门解决线性Diophantine系统。该方法已扩展到某些线性不等式的系统，以提供所有解决方案。在这里，我们主要集中在解决线性Diophantine系统和计算整数分解的秩降过程的理论研究上。研究了基本整数ABS算法，整数扩展ABS（IEABS）算法，一阶扰动问题，广义Rosser方法（GRA），扩展整数秩降阶过程，整数Wedderburn秩降阶公式以及相关的整数双共轭过程。我们提出了一个集成的整数秩降低过程，