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JEŚMANOWICZ’ CONJECTURE ON PYTHAGOREAN TRIPLES
Published online by Cambridge University Press: 13 March 2017
Abstract
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 $y\geq 2$.
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- © 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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