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JEŚMANOWICZ’ CONJECTURE ON PYTHAGOREAN TRIPLES

Part of: Diophantine equations

Published online by Cambridge University Press:  13 March 2017

MI-MI MA  and
YONG-GAO CHEN
MI-MI MA
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China email mamimi421@126.com
YONG-GAO CHEN*
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China email ygchen@njnu.edu.cn
*
✉ygchen@njnu.edu.cn
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Abstract

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In 1956, Jeśmanowicz conjectured that, for any positive integers $m$ and $n$ with $m>n$, $\gcd (m,n)=1$ and $2\nmid m+n$, the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ has only the positive integer solution $(x,y,z)=(2,2,2)$. In this paper, we prove the conjecture if $4\nmid mn$ and $y\geq 2$.

Keywords

Jeśmanowicz’ conjectureDiophantine equationPythagorean triple

MSC classification

Primary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 30 - 35
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported by the National Natural Science Foundation of China, grant no. 11371195, and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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