On Je\'{s}manowicz' Conjecture Concerning Pythagorean Triples
Received:August 26, 2014  Revised:November 24, 2014
Key Words: Je\'{s}manowicz' conjecture   Diophantine equation  
Fund Project:Supported by the Research Culture Fundation of Anhui Normal University (Grant Nos.2012xmpy009; 2014xmpy11) and the Natural Science Foundation of Anhui Province (Grant No.1208085QA02).
Author NameAffiliation
Cuifang SUN School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China 
Zhi CHENG School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China 
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Abstract:
      Let $(a,b,c)$ be a primitive Pythagorean triple. Je\'{s}manowicz conjectured in 1956 that for any positive integer $n$, the Diophantine equation $(an)^x+(bn)^y=(cn)^z$ has only the positive integer solution $(x,y,z)=(2,2,2)$. Let $p\equiv 3\pmod 4$ be a prime and $s$ be some positive integer. In the paper, we show that the conjecture is true when $(a,b,c)=(4p^{2s}-1,4p^{s},4p^{2s}+1)$ and certain divisibility conditions are satisfied.
Citation:
DOI:10.3770/j.issn:2095-2651.2015.02.004
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