On the Diophantine equation $(an)^{x}+(bn)^{y}=(cn)^{z}$
Bull. Korean Math. Soc. 2015 Vol. 52, No. 4, 1133-1138
https://doi.org/10.4134/BKMS.2015.52.4.1133
Published online July 31, 2015
Mi-Mi Ma and Jian-Dong Wu
Nanjing Normal University, Nanjing Normal University
Abstract : In 1956, Je\'{s}manowicz conjectured that, for any positive integer $n$ and any primitive Pythagorean triple $(a,b,c)$ with $a^2+b^2=c^2$, the equation $(an)^x+(bn)^y=(cn)^z$ has the unique solution $(x,y,z)=(2,2,2)$. In this paper, under some conditions, we prove the conjecture for the primitive Pythagorean triples $(a,b,c)=(4k^{2}-1,4k,4k^{2}+1)$.
Keywords : Je\'{s}manowicz' conjecture, Diophantine equation, Pythagorean triple
MSC numbers : 11D61
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