An Elementary Proof of a Result Ma and Chen

Qing Han; Pingzhi Yuan

doi:doi:10.11648/j.ajmcm.20200502.12

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An Elementary Proof of a Result Ma and Chen

Qing Han, Pingzhi Yuan

Published in American Journal of Mathematical and Computer Modelling (Volume 5, Issue 2)

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 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.

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

Keywords

Pythagorean Triple, Jesmanowicz Conjecture, Exponential Diophantine Equations

References

[1]	Z. Cao, A note on the Diophantine equation ax+by = cz, Acta Arith. 91 (1999) 85-93.
[2]	M. Deng, D. Huang, A note on Je_smanowicz’ conjecture concerning primitive Pythagorean triples. Bull. Aust. Math. Soc. 95 (2017) 5-13.
[3]	M. Deng, J. Guo, A note on Je_smanowicz’ conjecture concerning primitive Pythagorean triples. II. Acta Math. Hungar. 153 (2017) 436-448.
[4]	M. Deng and G. L. Cohen, On the conjecture of Je´smanowicz concerning Pythagorean triples, Bull. Aust. Math. Soc. 57 (1998) 515-524.
[5]	Y. Fujita and T. Miyazaki, Je_smanowicz’ conjecture with congruence relations, Colloq. Math. 128 (2012) 211-222.
[6]	Y. Guo and M. Le, A note on Je_smanowicz’ conjecture concerning Pythagorean numbers, Comment. Mat., Univ. St. Pauli 44 (1995) 225-228
[7]	Q. Han and P. Yuan, A note on Je_smanowicz’ conjecture, Acta Math Hungar. https://doi.org/10.1007/s10474-018- 0837-4.
[8]	L. Je_smanowicz, Several remarks on Pythagorean numbers. Wiadom. Mat. 1 (1955/56) 196-202.
[9]	M. Le, A note on Je_smanowicz’ conjecture concerning Pythagorean triples’, Bull. Aust. Math. Soc. 59 (1999), 477-480.
[10]	W. Lu, On the Pythagorean numbers 4n² - 1; 4n and 4n²+1, Acta Sci. Natur. Univ. Szechuan 2 (1959) 39-42.
[11]	Ma, Mi-Mi; Chen, Yong-Gao Je_smanowicz’ conjecture on Pythagorean triples. Bull. Aust. Math. Soc. 96 (2017) 30-35.
[12]	T. Miyazaki, On the conjecture of Jesmanowicz concerning Pythagorean triples, Bull. Aust. Math. Soc. 80 (2009), 413C422.
[13]	T. Miyazaki, Je_smanowicz’ conjecture on exponential Diophantine equations, Funct. Approximatio Comment Math. 45 (2011) 207-229.
[14]	T. Miyazaki, Generalizations of classical results on Je_smanowicz’ conjecture concerning Pythagorean triples, J. Number Theory 133 (2013) 583-595 .
[15]	T. Miyazaki, A remark on Je_smanowicz’ conjecture for the non-coprimality case, Acta Math. Sin. (Engl. Ser.) 31 (2015) 1255-1260.
[16]	T. Miyazaki, Contributions to some conjectures on a ternary exponential Diophantine equations, Acta Arith. (to appear).
[17]	T. Miyazaki and N. Terai, On Je_smanowicz’ conjecture concerning Pythagorean triples II, Acta Math. Hungar. 142 (2015) 286-293.
[18]	T. Miyazaki, P. Yuan, D. Wu, Generalizations of classical results on Je_smanowicz’ conjecture concerning Pythagorean triples II, J. Number Theory, 141 (2014) 184-201.
[19]	L. J. Mordell, Diophantine equations, London: Academic Press, 1969.
[20]	K. Takakuwa, A remark on Je_smanowicz’ conjecture, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996) 109- 110.
[21]	M. Tang, Z. Yang, Je_smanowicz’ conjecture revisited, Bull. Aust. Math. Soc. 88 (2013) 486-491.
[22]	N. Terai, Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations, Acta Arith. 90 (1999) 17-35.
[23]	N. Terai, On Je_emanowicz’ conjecture concerning primitive Pythagorean triples, J. Number Theory 141 (2014) 316-323.
[24]	P. Yuan and Q. Han, Je_smanowicz’ conjecture and related equations, Acta Arith. 184 (2018), 37-49.
[25]	X. Zhang, W. Zhang, The exponential Diophantine equation ((2^2m - 1)n)^x + (2^m+1n)^y = ((2^2m + 1)n)^z, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 57 (2014) 337-344.

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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://article.sciencepublishinggroup.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
N1  - https://doi.org/10.11648/j.ajmcm.20200502.12
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
SP  - 43
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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Author Information

Qing Han

School of Information Science and Technology, South China Business College of Guangdong University of Foreign Studies, Guangzhou, China
Pingzhi Yuan

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

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APA Style

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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ACS Style

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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AMA Style

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

Copy | Download

@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://article.sciencepublishinggroup.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}
}

Copy | Download

TY  - JOUR
T1  - An Elementary Proof of a Result Ma and Chen
AU  - Qing Han
AU  - Pingzhi Yuan
Y1  - 2020/04/23
PY  - 2020
N1  - https://doi.org/10.11648/j.ajmcm.20200502.12
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
SP  - 43
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  -

Copy | Download