| 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). |
|
| Hits: 2752 |
| Download times: 2897 |
| 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 |
| View Full Text View/Add Comment |
|
|
|
