Archive
Special Issues Volume 5, Issue 2, June 2020, Page: 43-46
An Elementary Proof of a Result Ma and Chen
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
Received: Jan. 15, 2020;       Accepted: Feb. 11, 2020;       Published: Apr. 23, 2020
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.
Keywords
Pythagorean Triple, Jesmanowicz Conjecture, Exponential Diophantine Equations
Qing Han, Pingzhi Yuan, An Elementary Proof of a Result Ma and Chen, American Journal of Mathematical and Computer Modelling. Vol. 5, No. 2, 2020, pp. 43-46. doi: 10.11648/j.ajmcm.20200502.12
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