On the diophantine equation Px + Qy = z²
Diophantine equation is a polynomial equation with two or more unknowns which integral solutions are required. An exponential Diophantine equation is an equation that has additional variable or variables occurring as exponents. This paper concentrates on finding an integral solution to the Diophanti...
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| Format: | Article |
| Language: | English |
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Faculty of Science and Natural Resources, Universiti Malaysia Sabah
2024
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| Online Access: | http://psasir.upm.edu.my/id/eprint/120155/ http://psasir.upm.edu.my/id/eprint/120155/1/120155.pdf |
| _version_ | 1848868125480583168 |
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| author | Japar, Izzati Izyani Sapar, Siti Hasana M. Johari, M. Aidil |
| author_facet | Japar, Izzati Izyani Sapar, Siti Hasana M. Johari, M. Aidil |
| author_sort | Japar, Izzati Izyani |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | Diophantine equation is a polynomial equation with two or more unknowns which integral solutions are required. An exponential Diophantine equation is an equation that has additional variable or variables occurring as exponents. This paper concentrates on finding an integral solution to the Diophantine equation px + qy = z2 for x + y = 5 and p, q are twin primes, cousin primes, sexy primes and also for any positive integers. By looking at the pattern of the solutions of each case, the theorems and lemmas will be constructed. In this paper, for all the possibilities of x + y = 5, it is proved that the Diophantine equation has no non-trivial solution for twin primes, cousin primes and sexy primes whereas for any positive integers, the equation has infinitely many solutions. |
| first_indexed | 2025-11-15T14:47:25Z |
| format | Article |
| id | upm-120155 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T14:47:25Z |
| publishDate | 2024 |
| publisher | Faculty of Science and Natural Resources, Universiti Malaysia Sabah |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-1201552025-09-24T03:25:28Z http://psasir.upm.edu.my/id/eprint/120155/ On the diophantine equation Px + Qy = z² Japar, Izzati Izyani Sapar, Siti Hasana M. Johari, M. Aidil Diophantine equation is a polynomial equation with two or more unknowns which integral solutions are required. An exponential Diophantine equation is an equation that has additional variable or variables occurring as exponents. This paper concentrates on finding an integral solution to the Diophantine equation px + qy = z2 for x + y = 5 and p, q are twin primes, cousin primes, sexy primes and also for any positive integers. By looking at the pattern of the solutions of each case, the theorems and lemmas will be constructed. In this paper, for all the possibilities of x + y = 5, it is proved that the Diophantine equation has no non-trivial solution for twin primes, cousin primes and sexy primes whereas for any positive integers, the equation has infinitely many solutions. Faculty of Science and Natural Resources, Universiti Malaysia Sabah 2024 Article PeerReviewed text en http://psasir.upm.edu.my/id/eprint/120155/1/120155.pdf Japar, Izzati Izyani and Sapar, Siti Hasana and M. Johari, M. Aidil (2024) On the diophantine equation Px + Qy = z². Borneo Science (The Journal of Science and Technology), 45 (2). pp. 1-10. ISSN 1394-4339; eISSN: 2231-9085 https://jurcon.ums.edu.my/ojums/index.php/borneo-science/article/view/5878 10.51200/bsj.v45i2.5878 |
| spellingShingle | Japar, Izzati Izyani Sapar, Siti Hasana M. Johari, M. Aidil On the diophantine equation Px + Qy = z² |
| title | On the diophantine equation Px + Qy = z² |
| title_full | On the diophantine equation Px + Qy = z² |
| title_fullStr | On the diophantine equation Px + Qy = z² |
| title_full_unstemmed | On the diophantine equation Px + Qy = z² |
| title_short | On the diophantine equation Px + Qy = z² |
| title_sort | on the diophantine equation px + qy = z² |
| url | http://psasir.upm.edu.my/id/eprint/120155/ http://psasir.upm.edu.my/id/eprint/120155/ http://psasir.upm.edu.my/id/eprint/120155/ http://psasir.upm.edu.my/id/eprint/120155/1/120155.pdf |