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Methods of Solving Diophantine Equations in Secondary Education in Romania

Received: 8 February 2014     Published: 10 March 2014
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Abstract

This study aims to highlight the importance of knowing the methods of solving Diophantine equations. The material is structured into: Introduction, Classes of Diophantine equations, presentation of first-degree Diophantine equations, Pythagorean triples and higher - Diophantine equations, methods for solving Diophantine equations. The paper describes and exemplifies different methods such as the decomposition method, the parametric method for solving Diophantine equations, solving Diophantine equations with inequalities through the method of modular arithmetic, mathematical induction, Fermat's method of infinite descent. Solving problems is illustrated by various applications of the mathematical results methods presented above. Any education, including mathematical education, has a double effect. On the one hand, the learner gains knowledge, on the other hand, he builds those skills which are engaged in work, develop the abilities needed to perform this education. Mathematical education builds thought. Of course, other actions are involved in building thought, but the role of Mathematical education is essential. This article is part of an empirical research on the teaching and learning of mathematics, teaching practices related to the main classes of Diophantine equations, leading to the development of cognitive skills in students.

Published in Science Journal of Education (Volume 2, Issue 1)
DOI 10.11648/j.sjedu.20140201.14
Page(s) 22-32
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Diophantine Equations, Linear Equations, Pythagorean Equations, Bachet Equations

References
[1] T. Andreescu, D. Andrica, „An introduction to the study of equations diofantiene", GIL Publishing House, Timisoara, 2002.
[2] A. Balaucă, „Arithmetic, Algebra, Geometry, 850 problem Olympiads and competitions , Class VI", Taida Publishing House , Iasi, 2004.
[3] D. Constantinescu , P. Dumitrescu „Mathematics problems, Class VI", House "OFFEST COLOR", Ramnicu Valcea, 2000.
[4] I. Cucurezeanu, „Issues of arithmetic and number theory", Technical Publishing, 1976.
[5] D. Lenț , M. Linț, D. S. Marinescu, R. Marinescu, „Performance in mathematics topics and issues to prepare for competitions, cl . VII - VIII", Corvin Publishing House, Deva, 2005.
[6] C. Lupu, „Arithmetic. Theory. Problems. Solving Methods", Draw Publishing House, Bacau, 2003.
[7] C. Lupu „Use Of Mathematical Content Sheets", Procedia - Social and Behavioural Sciences, Volum 93 ( 2013 ), pp. 1673 – 1680.
[8] S. Peligrad, D. Zaharia, S. Simion, M. Zaharia, „Arithmetic , Algebra, Geometry", Parallel 42 House, Mate Collection, 2000.
[9] M.M. Postnikov, „On Fermat's theorem", Didactic and Pedagogic Publishing House , Bucharest, 1983.
[10] W. Sierpinski - „What do we know and what we do not know about prime numbers", Scientific Publishing, 1966.
[11] Collection „Mathematical Gazette" , 1990-2009 .
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    Lupu Costică. (2014). Methods of Solving Diophantine Equations in Secondary Education in Romania. Science Journal of Education, 2(1), 22-32. https://doi.org/10.11648/j.sjedu.20140201.14

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    ACS Style

    Lupu Costică. Methods of Solving Diophantine Equations in Secondary Education in Romania. Sci. J. Educ. 2014, 2(1), 22-32. doi: 10.11648/j.sjedu.20140201.14

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    AMA Style

    Lupu Costică. Methods of Solving Diophantine Equations in Secondary Education in Romania. Sci J Educ. 2014;2(1):22-32. doi: 10.11648/j.sjedu.20140201.14

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  • @article{10.11648/j.sjedu.20140201.14,
      author = {Lupu Costică},
      title = {Methods of Solving Diophantine Equations in Secondary Education in Romania},
      journal = {Science Journal of Education},
      volume = {2},
      number = {1},
      pages = {22-32},
      doi = {10.11648/j.sjedu.20140201.14},
      url = {https://doi.org/10.11648/j.sjedu.20140201.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjedu.20140201.14},
      abstract = {This study aims to highlight the importance of knowing the methods of solving  Diophantine equations. The material is structured into: Introduction, Classes of Diophantine equations, presentation of first-degree Diophantine equations, Pythagorean triples and higher - Diophantine equations, methods for solving Diophantine equations. The paper describes and exemplifies different methods such as the decomposition method, the parametric method for solving Diophantine equations, solving Diophantine equations with inequalities through the method of modular arithmetic, mathematical induction, Fermat's method of infinite descent. Solving problems is illustrated by various applications of the mathematical results methods presented above. Any education, including mathematical education, has a double effect. On the one hand, the learner gains knowledge, on the other hand, he builds those skills which are engaged in work, develop the abilities needed to perform this education. Mathematical education builds thought. Of course, other actions are involved in building thought, but the role of Mathematical education is essential. This article is part of an empirical research on the teaching and learning of mathematics, teaching practices related to the main classes of Diophantine equations, leading to the development of cognitive skills in students.},
     year = {2014}
    }
    

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    AB  - This study aims to highlight the importance of knowing the methods of solving  Diophantine equations. The material is structured into: Introduction, Classes of Diophantine equations, presentation of first-degree Diophantine equations, Pythagorean triples and higher - Diophantine equations, methods for solving Diophantine equations. The paper describes and exemplifies different methods such as the decomposition method, the parametric method for solving Diophantine equations, solving Diophantine equations with inequalities through the method of modular arithmetic, mathematical induction, Fermat's method of infinite descent. Solving problems is illustrated by various applications of the mathematical results methods presented above. Any education, including mathematical education, has a double effect. On the one hand, the learner gains knowledge, on the other hand, he builds those skills which are engaged in work, develop the abilities needed to perform this education. Mathematical education builds thought. Of course, other actions are involved in building thought, but the role of Mathematical education is essential. This article is part of an empirical research on the teaching and learning of mathematics, teaching practices related to the main classes of Diophantine equations, leading to the development of cognitive skills in students.
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