Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $
In this paper, we consider the simultaneous Pell equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equatio...
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AIMS Press
2023-06-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023987?viewType=HTML |
| _version_ | 1827915303096418304 |
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| author | Cencen Dou Jiagui Luo |
| author_facet | Cencen Dou Jiagui Luo |
| author_sort | Cencen Dou |
| collection | DOAJ |
| description | In this paper, we consider the simultaneous Pell equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtain 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 in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $ (a^2+2)x^2-y^2 = 2 $ 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 | 2024-03-13T02:57:34Z |
| format | Article |
| id | doaj.art-66125766d18242bb9fe936c9001ec7e2 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-13T02:57:34Z |
| publishDate | 2023-06-01 |
| publisher | AIMS Press |
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| spelling | doaj.art-66125766d18242bb9fe936c9001ec7e22023-06-28T00:59:14ZengAIMS PressAIMS Mathematics2473-69882023-06-0188193531937310.3934/math.2023987Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $Cencen Dou0Jiagui 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+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtain 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 in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $ (a^2+2)x^2-y^2 = 2 $ 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://www.aimspress.com/article/doi/10.3934/math.2023987?viewType=HTMLdiophantine equationssimultaneous pell equationsminimal solutionslehmer sequences |
| spellingShingle | Cencen Dou Jiagui Luo Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ AIMS Mathematics diophantine equations simultaneous pell equations minimal solutions lehmer sequences |
| title | Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ |
| title_full | Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ |
| title_fullStr | Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ |
| title_full_unstemmed | Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ |
| title_short | Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ |
| title_sort | complete solutions of the simultaneous pell s equations a 2 2 x 2 y 2 2 and x 2 bz 2 1 |
| topic | diophantine equations simultaneous pell equations minimal solutions lehmer sequences |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023987?viewType=HTML |
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