Further results on the Diophantine equation x^2+16·7^b = y^n when n is even
This work extends the results for the Diophantine equation x^2+16∙7^b = y^n for n=2r, where x, y, b, r ∈ Z^+. Earlier results classified the generators of solutions, which are the pair of integers (x, y^r), into several categories and presented the general formula that determines the values of x and...
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| Format: | Article |
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Prince of Songkla University
2024
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| Online Access: | http://psasir.upm.edu.my/id/eprint/118877/ |
| _version_ | 1848867811709943808 |
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| author | Yow, Kai Siong Sapar, Siti Hasana Pham, Hoa |
| author_facet | Yow, Kai Siong Sapar, Siti Hasana Pham, Hoa |
| author_sort | Yow, Kai Siong |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | This work extends the results for the Diophantine equation x^2+16∙7^b = y^n for n=2r, where x, y, b, r ∈ Z^+. Earlier results classified the generators of solutions, which are the pair of integers (x, y^r), into several categories and presented the general formula that determines the values of x and y^r for the respective category. The lower bound for the number of non-negative integral solutions associated with each b is also provided. We now extend the results and prove the necessary and sufficient conditions required to obtain integral solutions x and y to the equation, by considering various scenarios based on the parity of b. We also determine the values of n in which integral solutions exist. |
| first_indexed | 2025-11-15T14:42:26Z |
| format | Article |
| id | upm-118877 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-15T14:42:26Z |
| publishDate | 2024 |
| publisher | Prince of Songkla University |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-1188772025-07-28T07:02:37Z http://psasir.upm.edu.my/id/eprint/118877/ Further results on the Diophantine equation x^2+16·7^b = y^n when n is even Yow, Kai Siong Sapar, Siti Hasana Pham, Hoa This work extends the results for the Diophantine equation x^2+16∙7^b = y^n for n=2r, where x, y, b, r ∈ Z^+. Earlier results classified the generators of solutions, which are the pair of integers (x, y^r), into several categories and presented the general formula that determines the values of x and y^r for the respective category. The lower bound for the number of non-negative integral solutions associated with each b is also provided. We now extend the results and prove the necessary and sufficient conditions required to obtain integral solutions x and y to the equation, by considering various scenarios based on the parity of b. We also determine the values of n in which integral solutions exist. Prince of Songkla University 2024 Article PeerReviewed Yow, Kai Siong and Sapar, Siti Hasana and Pham, Hoa (2024) Further results on the Diophantine equation x^2+16·7^b = y^n when n is even. Songklanakarin Journal of Science and Technology, 46 (3). pp. 294-301. ISSN 0125-3395 https://sjst.psu.ac.th/journal/46-3/7.pdf |
| spellingShingle | Yow, Kai Siong Sapar, Siti Hasana Pham, Hoa Further results on the Diophantine equation x^2+16·7^b = y^n when n is even |
| title | Further results on the Diophantine equation x^2+16·7^b = y^n when n is even |
| title_full | Further results on the Diophantine equation x^2+16·7^b = y^n when n is even |
| title_fullStr | Further results on the Diophantine equation x^2+16·7^b = y^n when n is even |
| title_full_unstemmed | Further results on the Diophantine equation x^2+16·7^b = y^n when n is even |
| title_short | Further results on the Diophantine equation x^2+16·7^b = y^n when n is even |
| title_sort | further results on the diophantine equation x^2+16·7^b = y^n when n is even |
| url | http://psasir.upm.edu.my/id/eprint/118877/ http://psasir.upm.edu.my/id/eprint/118877/ |