Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $
In this paper, we consider the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ where $ a > 0 $ is an integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtained the necessary and sufficient conditions that the above simultaneous Pell equations ha...
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AIMS Press
2021-07-01
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| Online Access: | https://aimspress.com/article/doi/10.3934/math.2021577?viewType=HTML |
| _version_ | 1818741419524751360 |
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| author | Changsheng Luo Jiagui Luo |
| author_facet | Changsheng Luo Jiagui Luo |
| author_sort | Changsheng Luo |
| collection | DOAJ |
| description | In this paper, we consider the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ where $ a > 0 $ is an integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtained the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution. When a solution exists, assuming the positive solutions of the Pell equation $ x^2(a^2+1)-y^2 = -1 $ are $ x = x_m $ and $ y = y_m $ with $ m\geq 1 $ odd, then the only solution of the system is given by $ m = 3 $ or $ m = 5 $ or $ m = 7 $ or $ m = 9 $. |
| first_indexed | 2024-12-18T01:56:19Z |
| format | Article |
| id | doaj.art-2208dfd22cb245e29274ecf0e53a0e8f |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-18T01:56:19Z |
| publishDate | 2021-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-2208dfd22cb245e29274ecf0e53a0e8f2022-12-21T21:24:53ZengAIMS PressAIMS Mathematics2473-69882021-07-01699919993810.3934/math.2021577Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $Changsheng Luo0Jiagui Luo1School of Mathematics and Information, China West Normal University, Nanchong 637009, ChinaSchool of Mathematics and Information, China West Normal University, Nanchong 637009, ChinaIn this paper, we consider the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ where $ a > 0 $ is an integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtained the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution. When a solution exists, assuming the positive solutions of the Pell equation $ x^2(a^2+1)-y^2 = -1 $ are $ x = x_m $ and $ y = y_m $ with $ m\geq 1 $ odd, then the only solution of the system is given by $ m = 3 $ or $ m = 5 $ or $ m = 7 $ or $ m = 9 $.https://aimspress.com/article/doi/10.3934/math.2021577?viewType=HTMLdiophantine equationssimultaneous pell equationsminimal solutionslucas sequences |
| spellingShingle | Changsheng Luo Jiagui Luo Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ AIMS Mathematics diophantine equations simultaneous pell equations minimal solutions lucas sequences |
| title | Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ |
| title_full | Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ |
| title_fullStr | Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ |
| title_full_unstemmed | Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ |
| title_short | Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ |
| title_sort | complete solutions of the simultaneous pell equations a 2 1 y 2 x 2 y 2 bz 2 1 |
| topic | diophantine equations simultaneous pell equations minimal solutions lucas sequences |
| url | https://aimspress.com/article/doi/10.3934/math.2021577?viewType=HTML |
| work_keys_str_mv | AT changshengluo completesolutionsofthesimultaneouspellequationsa21y2x2y2bz21 AT jiaguiluo completesolutionsofthesimultaneouspellequationsa21y2x2y2bz21 |