Further results on the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ when ๐ is even
This work extends the results for the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ for ๐ = 2๐, where ๐ฅ, ๐ฆ, ๐, ๐ โ โค + . Earlier results classified the generators of solutions, which are the pair of integers (๐ฅ, ๐ฆ ๐ ), into several categories and presented the general formula that determines t...
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| Format: | Article |
| Language: | English |
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Prince of Songkla University
2024-06-01
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| Series: | Songklanakarin Journal of Science and Technology (SJST) |
| Subjects: | |
| Online Access: | https://sjst.psu.ac.th/journal/46-3/7.pdf |
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| _version_ | 1849719026042798080 |
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| author | Siti Hasana Sapar Kai Siong Yow Hoa Pham |
| author_facet | Siti Hasana Sapar Kai Siong Yow Hoa Pham |
| author_sort | Siti Hasana Sapar |
| collection | DOAJ |
| description | This work extends the results for the Diophantine equation ๐ฅ
2 + 16 โ 7
๐ = ๐ฆ
๐
for ๐ = 2๐, where ๐ฅ, ๐ฆ, ๐, ๐ โ โค
+
. Earlier
results classified the generators of solutions, which are the pair of integers (๐ฅ, ๐ฆ
๐
), into several categories and presented the general
formula that determines the values of ๐ฅ and ๐ฆ
๐
for the respective category. The lower bound for the number of non-negative integral
solutions associated with each ๐ is also provided. We now extend the results and prove the necessary and sufficient conditions
required to obtain integral solutions ๐ฅ and ๐ฆ to the equation, by considering various scenarios based on the parity of ๐. We also
determine the values of ๐ for which integral solutions exist.
|
| format | Article |
| id | doaj-art-1e5aecaf1b9841cb9b66cdd6a9941e48 |
| institution | DOAJ |
| issn | 0125-3395 |
| language | English |
| publishDate | 2024-06-01 |
| publisher | Prince of Songkla University |
| record_format | Article |
| series | Songklanakarin Journal of Science and Technology (SJST) |
| spelling | doaj-art-1e5aecaf1b9841cb9b66cdd6a9941e482025-08-20T03:12:14ZengPrince of Songkla UniversitySongklanakarin Journal of Science and Technology (SJST)0125-33952024-06-01463294301Further results on the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ when ๐ is evenSiti Hasana Sapar0Kai Siong Yow1Hoa Pham2Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang, Selangor, 43400 MalaysiaDepartment of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang, Selangor, 43400 MalaysiaDepartment of Mathematics, An Giang University, Vietnam National University, Ho Chi Minh City, VietnamThis work extends the results for the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ for ๐ = 2๐, where ๐ฅ, ๐ฆ, ๐, ๐ โ โค + . Earlier results classified the generators of solutions, which are the pair of integers (๐ฅ, ๐ฆ ๐ ), into several categories and presented the general formula that determines the values of ๐ฅ and ๐ฆ ๐ for the respective category. The lower bound for the number of non-negative integral solutions associated with each ๐ is also provided. We now extend the results and prove the necessary and sufficient conditions required to obtain integral solutions ๐ฅ and ๐ฆ to the equation, by considering various scenarios based on the parity of ๐. We also determine the values of ๐ for which integral solutions exist. https://sjst.psu.ac.th/journal/46-3/7.pdfdiophantine equationpolynomialgeneratorintegral solution |
| spellingShingle | Siti Hasana Sapar Kai Siong Yow Hoa Pham Further results on the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ when ๐ is even Songklanakarin Journal of Science and Technology (SJST) diophantine equation polynomial generator integral solution |
| title | Further results on the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ when ๐ is even |
| title_full | Further results on the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ when ๐ is even |
| title_fullStr | Further results on the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ when ๐ is even |
| title_full_unstemmed | Further results on the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ when ๐ is even |
| title_short | Further results on the Diophantine equation ๐ฅ 2 + 16 โ 7 ๐ = ๐ฆ ๐ when ๐ is even |
| title_sort | further results on the diophantine equation ๐ฅ 2 16 โ 7 ๐ ๐ฆ ๐ when ๐ is even |
| topic | diophantine equation polynomial generator integral solution |
| url | https://sjst.psu.ac.th/journal/46-3/7.pdf |
| work_keys_str_mv | AT sitihasanasapar furtherresultsonthediophantineequationx2167bynwhenniseven AT kaisiongyow furtherresultsonthediophantineequationx2167bynwhenniseven AT hoapham furtherresultsonthediophantineequationx2167bynwhenniseven |