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Rank Reduction Processes for Solving Linear Diophantine Systems and Integer Factorizations: A Review

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Abstract

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.

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Acknowledgements

The first author thanks the Research Council of Qom University and the second author thanks the Research Council of Sharif University of Technology for supporting this work.

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Authors and Affiliations

  1. Department of Mathematics, University of Qom, Qom, Iran

    Effat Golpar-Raboky & Nezam Mahdavi-Amiri

  2. Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Effat Golpar-Raboky & Nezam Mahdavi-Amiri

Authors
  1. Effat Golpar-Raboky
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  2. Nezam Mahdavi-Amiri
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Correspondence to Effat Golpar-Raboky.

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Communicated by Davod Khojasteh Salkuyeh.

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Golpar-Raboky, E., Mahdavi-Amiri, N. Rank Reduction Processes for Solving Linear Diophantine Systems and Integer Factorizations: A Review. Bull. Iran. Math. Soc. 46, 647–667 (2020). https://doi.org/10.1007/s41980-019-00282-8

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  • DOI: https://doi.org/10.1007/s41980-019-00282-8

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