American Journal of Mathematical and Computer Modelling

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An Elementary Proof of a Result Ma and Chen

Received: 15 January 2020    Accepted: 11 February 2020    Published: 23 April 2020
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Abstract

In 1956, Je_smanowicz conjectured that, for positive integers m and n with m > n; gcd(m; n) = 1 and mn (mod 2), the exponential Diophantine equation (m2 -n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution (x; y; z) = (2; 2; 2). Recently, Ma and Chen proved the conjecture if 4 mn and y ≥ 2. In this paper, we provide a proposition that, for positive integers m and n with m > n; gcd(m; n) = 1 and m2 + n2 ≡ 5 (mod 8), the exponential Diophantine equation (m2 -n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution x = y = z = 2 with 2j gcd(x; y). Then we present an elementary and simple proof of the result of Ma and Chen by using Jacobi’s symbols.

DOI 10.11648/j.ajmcm.20200502.12
Published in American Journal of Mathematical and Computer Modelling (Volume 5, Issue 2, June 2020)
Page(s) 43-46
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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), 2024. Published by Science Publishing Group

Keywords

Pythagorean Triple, Jesmanowicz Conjecture, Exponential Diophantine Equations

References
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Author Information
  • School of Information Science and Technology, South China Business College of Guangdong University of Foreign Studies, Guangzhou, China

  • School of Mathematics, South China Normal University, Guangzhou, China

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    Qing Han, Pingzhi Yuan. (2020). An Elementary Proof of a Result Ma and Chen. American Journal of Mathematical and Computer Modelling, 5(2), 43-46. https://doi.org/10.11648/j.ajmcm.20200502.12

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    Qing Han; Pingzhi Yuan. An Elementary Proof of a Result Ma and Chen. Am. J. Math. Comput. Model. 2020, 5(2), 43-46. doi: 10.11648/j.ajmcm.20200502.12

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    Qing Han, Pingzhi Yuan. An Elementary Proof of a Result Ma and Chen. Am J Math Comput Model. 2020;5(2):43-46. doi: 10.11648/j.ajmcm.20200502.12

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  • @article{10.11648/j.ajmcm.20200502.12,
      author = {Qing Han and Pingzhi Yuan},
      title = {An Elementary Proof of a Result Ma and Chen},
      journal = {American Journal of Mathematical and Computer Modelling},
      volume = {5},
      number = {2},
      pages = {43-46},
      doi = {10.11648/j.ajmcm.20200502.12},
      url = {https://doi.org/10.11648/j.ajmcm.20200502.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajmcm.20200502.12},
      abstract = {In 1956, Je_smanowicz conjectured that, for positive integers m and n with m > n; gcd(m; n) = 1 and m ≢ n (mod 2), the exponential Diophantine equation (m2 -n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution (x; y; z) = (2; 2; 2). Recently, Ma and Chen proved the conjecture if 4 ☓mn and y ≥ 2. In this paper, we provide a proposition that, for positive integers m and n with m > n; gcd(m; n) = 1 and m2 + n2 ≡ 5 (mod 8), the exponential Diophantine equation (m2 -n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution x = y = z = 2 with 2j gcd(x; y). Then we present an elementary and simple proof of the result of Ma and Chen by using Jacobi’s symbols.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - An Elementary Proof of a Result Ma and Chen
    AU  - Qing Han
    AU  - Pingzhi Yuan
    Y1  - 2020/04/23
    PY  - 2020
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    DO  - 10.11648/j.ajmcm.20200502.12
    T2  - American Journal of Mathematical and Computer Modelling
    JF  - American Journal of Mathematical and Computer Modelling
    JO  - American Journal of Mathematical and Computer Modelling
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    EP  - 46
    PB  - Science Publishing Group
    SN  - 2578-8280
    UR  - https://doi.org/10.11648/j.ajmcm.20200502.12
    AB  - In 1956, Je_smanowicz conjectured that, for positive integers m and n with m > n; gcd(m; n) = 1 and m ≢ n (mod 2), the exponential Diophantine equation (m2 -n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution (x; y; z) = (2; 2; 2). Recently, Ma and Chen proved the conjecture if 4 ☓mn and y ≥ 2. In this paper, we provide a proposition that, for positive integers m and n with m > n; gcd(m; n) = 1 and m2 + n2 ≡ 5 (mod 8), the exponential Diophantine equation (m2 -n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution x = y = z = 2 with 2j gcd(x; y). Then we present an elementary and simple proof of the result of Ma and Chen by using Jacobi’s symbols.
    VL  - 5
    IS  - 2
    ER  - 

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