On the Diophantine Equation 5 x + p mn y = z 2
Diophantine equation is a polynomial equation with two or more unknowns for which only integral solutions are sought. This paper concentrates on finding the integral solutions to the Diophantine equation 5 x + p mn y = z 2 where p > 5 a prime number and y = 1, 2. The positive integral solutions t...
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| Format: | Article |
| Language: | English |
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Institute for Mathematical Research, Universiti Putra Malaysia
2019
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| Online Access: | http://psasir.upm.edu.my/id/eprint/81535/ http://psasir.upm.edu.my/id/eprint/81535/1/Diophantine.pdf |
| _version_ | 1848859127729618944 |
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| author | Bakar, H. S. Sapar, S. H. Johari, M. A. M. |
| author_facet | Bakar, H. S. Sapar, S. H. Johari, M. A. M. |
| author_sort | Bakar, H. S. |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | Diophantine equation is a polynomial equation with two or more unknowns for which only integral solutions are sought. This paper concentrates on finding the integral solutions to the Diophantine equation 5 x + p mn y = z 2 where p > 5 a prime number and y = 1, 2. The positive integral solutions to the equation are (x, m, n, y, z) = (2r, t, pt k 2 ± 2k5 r , 1, pt k ± 5 r ) and 2r, 2t, 5 2r−α − 5 α 2p t , 2, 5 2r−α + 5α 2 for k, r, t ∈ N where 0 ≤ α < r. |
| first_indexed | 2025-11-15T12:24:24Z |
| format | Article |
| id | upm-81535 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T12:24:24Z |
| publishDate | 2019 |
| publisher | Institute for Mathematical Research, Universiti Putra Malaysia |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-815352024-05-14T07:13:23Z http://psasir.upm.edu.my/id/eprint/81535/ On the Diophantine Equation 5 x + p mn y = z 2 Bakar, H. S. Sapar, S. H. Johari, M. A. M. Diophantine equation is a polynomial equation with two or more unknowns for which only integral solutions are sought. This paper concentrates on finding the integral solutions to the Diophantine equation 5 x + p mn y = z 2 where p > 5 a prime number and y = 1, 2. The positive integral solutions to the equation are (x, m, n, y, z) = (2r, t, pt k 2 ± 2k5 r , 1, pt k ± 5 r ) and 2r, 2t, 5 2r−α − 5 α 2p t , 2, 5 2r−α + 5α 2 for k, r, t ∈ N where 0 ≤ α < r. Institute for Mathematical Research, Universiti Putra Malaysia 2019 Article PeerReviewed text en http://psasir.upm.edu.my/id/eprint/81535/1/Diophantine.pdf Bakar, H. S. and Sapar, S. H. and Johari, M. A. M. (2019) On the Diophantine Equation 5 x + p mn y = z 2. Malaysian Journal of Mathematical Sciences, 13 (spec.). pp. 41-50. ISSN 1823-8343; ESSN: 2289-750X https://mjms.upm.edu.my/senaraimakalah.php?yr=2019&bln=April&vol=13(S) |
| spellingShingle | Bakar, H. S. Sapar, S. H. Johari, M. A. M. On the Diophantine Equation 5 x + p mn y = z 2 |
| title | On the Diophantine Equation 5 x + p mn y = z 2 |
| title_full | On the Diophantine Equation 5 x + p mn y = z 2 |
| title_fullStr | On the Diophantine Equation 5 x + p mn y = z 2 |
| title_full_unstemmed | On the Diophantine Equation 5 x + p mn y = z 2 |
| title_short | On the Diophantine Equation 5 x + p mn y = z 2 |
| title_sort | on the diophantine equation 5 x + p mn y = z 2 |
| url | http://psasir.upm.edu.my/id/eprint/81535/ http://psasir.upm.edu.my/id/eprint/81535/ http://psasir.upm.edu.my/id/eprint/81535/1/Diophantine.pdf |