The Exponential Diophantine Equation 2x+by=cz
Let b and c be fixed coprime odd positive integers with min{b,c}>1. In this paper, a classification of all positive integer solutions (x,y,z) of the equation 2x+by=cz is given. Further, by an elementary approach, we prove that if c=b+2, then the equation has only the positive integer solution (x,...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/401816 |
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| _version_ | 1832558160366272512 |
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| author | Yahui Yu Xiaoxue Li |
| author_facet | Yahui Yu Xiaoxue Li |
| author_sort | Yahui Yu |
| collection | DOAJ |
| description | Let b and c be fixed coprime odd positive integers with min{b,c}>1. In this paper, a classification of all positive integer solutions (x,y,z) of the equation 2x+by=cz is given. Further, by an elementary approach, we prove that if c=b+2, then the equation has only the positive integer solution (x,y,z)=(1,1,1), except for (b,x,y,z)=(89,13,1,2) and (2r-1,r+2,2,2), where r is a positive integer with r≥2. |
| format | Article |
| id | doaj-art-26ed904df459404286c818a9e49a41f0 |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-26ed904df459404286c818a9e49a41f02025-02-03T01:33:07ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/401816401816The Exponential Diophantine Equation 2x+by=czYahui Yu0Xiaoxue Li1Luoyang Institute of Science and Technology, Luoyang, Henan 471023, ChinaSchool of Mathematics, Northwest University, Xi'an, Shaanxi 710127, ChinaLet b and c be fixed coprime odd positive integers with min{b,c}>1. In this paper, a classification of all positive integer solutions (x,y,z) of the equation 2x+by=cz is given. Further, by an elementary approach, we prove that if c=b+2, then the equation has only the positive integer solution (x,y,z)=(1,1,1), except for (b,x,y,z)=(89,13,1,2) and (2r-1,r+2,2,2), where r is a positive integer with r≥2.http://dx.doi.org/10.1155/2014/401816 |
| spellingShingle | Yahui Yu Xiaoxue Li The Exponential Diophantine Equation 2x+by=cz The Scientific World Journal |
| title | The Exponential Diophantine Equation 2x+by=cz |
| title_full | The Exponential Diophantine Equation 2x+by=cz |
| title_fullStr | The Exponential Diophantine Equation 2x+by=cz |
| title_full_unstemmed | The Exponential Diophantine Equation 2x+by=cz |
| title_short | The Exponential Diophantine Equation 2x+by=cz |
| title_sort | exponential diophantine equation 2x by cz |
| url | http://dx.doi.org/10.1155/2014/401816 |
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