Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1
Pell equation is a special type of Diophantine equations of the form x2 − my2 = 1, where m is a positive non-square integer. Since m is not a perfect square, then there exist infinitely many integer solutions (x, y) to the Pell equation. This paper will discuss the integral solutions to the simultan...
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| Format: | Article |
| Language: | English |
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Institute for Mathematical Research, Universiti Putra Malaysia
2017
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| Online Access: | http://psasir.upm.edu.my/id/eprint/51916/ http://psasir.upm.edu.my/id/eprint/51916/1/6.%20Amiranhasana.pdf |
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| author | Sihabudin, Nurul Amirah Sapar, Siti Hasana Mohamad Johari, Mohamad Aidil |
| author_facet | Sihabudin, Nurul Amirah Sapar, Siti Hasana Mohamad Johari, Mohamad Aidil |
| author_sort | Sihabudin, Nurul Amirah |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | Pell equation is a special type of Diophantine equations of the form x2 − my2 = 1, where m is a positive non-square integer. Since m is not a perfect square, then there exist infinitely many integer solutions (x, y) to the Pell equation. This paper will discuss the integral solutions to the simultaneous Pell equations x2 − my2 = 1 and y2 − pz2 = 1, where m is square free integer and p is odd prime. The solutions of these simultaneous equations are of the form of (x, y, z, m) = (yn2t±1, yn, zn, yn2t2±2t) and (y2n/2 t ±1, yn, zn, y2n/4 t2) for yn odd and even respectively, where t ∈ N. |
| first_indexed | 2025-11-15T10:30:32Z |
| format | Article |
| id | upm-51916 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T10:30:32Z |
| publishDate | 2017 |
| publisher | Institute for Mathematical Research, Universiti Putra Malaysia |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-519162017-05-03T04:34:42Z http://psasir.upm.edu.my/id/eprint/51916/ Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1 Sihabudin, Nurul Amirah Sapar, Siti Hasana Mohamad Johari, Mohamad Aidil Pell equation is a special type of Diophantine equations of the form x2 − my2 = 1, where m is a positive non-square integer. Since m is not a perfect square, then there exist infinitely many integer solutions (x, y) to the Pell equation. This paper will discuss the integral solutions to the simultaneous Pell equations x2 − my2 = 1 and y2 − pz2 = 1, where m is square free integer and p is odd prime. The solutions of these simultaneous equations are of the form of (x, y, z, m) = (yn2t±1, yn, zn, yn2t2±2t) and (y2n/2 t ±1, yn, zn, y2n/4 t2) for yn odd and even respectively, where t ∈ N. Institute for Mathematical Research, Universiti Putra Malaysia 2017 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/51916/1/6.%20Amiranhasana.pdf Sihabudin, Nurul Amirah and Sapar, Siti Hasana and Mohamad Johari, Mohamad Aidil (2017) Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1. Malaysian Journal of Mathematical Sciences, 11 (spec. Apr.). pp. 61-71. ISSN 1823-8343; ESSN: 2289-750X http://einspem.upm.edu.my/journal/fullpaper/vol11sapril/6.%20Amiranhasana.pdf |
| spellingShingle | Sihabudin, Nurul Amirah Sapar, Siti Hasana Mohamad Johari, Mohamad Aidil Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1 |
| title | Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1 |
| title_full | Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1 |
| title_fullStr | Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1 |
| title_full_unstemmed | Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1 |
| title_short | Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1 |
| title_sort | simultaneous pell equations x2 - my2 = 1 and y2 - pz2 = 1 |
| url | http://psasir.upm.edu.my/id/eprint/51916/ http://psasir.upm.edu.my/id/eprint/51916/ http://psasir.upm.edu.my/id/eprint/51916/1/6.%20Amiranhasana.pdf |