Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ
We present a method of determining integral solutions to the equation x2 + 16 ∙ 7b = y2r, where x, y, b, r ∈ ℤ+. We observe that the results can be classified into several categories. Under each category, a general formula is obtained using the geometric progression method. We then provide the bound...
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sg-ntu-dr.10356-1645232023-01-30T08:12:51Z Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ Yow, Kai Siong Sapar, Siti Hasana Low, Cheng Yaw School of Computer Science and Engineering Engineering::Computer science and engineering Diophantine Equation Integral Solution We present a method of determining integral solutions to the equation x2 + 16 ∙ 7b = y2r, where x, y, b, r ∈ ℤ+. We observe that the results can be classified into several categories. Under each category, a general formula is obtained using the geometric progression method. We then provide the bound for the number of non-negative integral solutions associated with each b. Lastly, the general formula for each of the categories is obtained and presented to determine the respective values of x and yr. We also highlight two special cases where different formulae are needed to represent their integral solutions. National Research Foundation (NRF) Published version This research was partially supported by Universiti Putra Malaysia under Putra Grant GP-IPM/2020/9684300and National Research Foundation Singapore. 2023-01-30T08:12:50Z 2023-01-30T08:12:50Z 2022 Journal Article Yow, K. S., Sapar, S. H. & Low, C. Y. (2022). Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ. Malaysian Journal of Fundamental and Applied Sciences, 18(4), 489-496. https://dx.doi.org/10.11113/mjfas.v18n4.2580 2289-599X https://hdl.handle.net/10356/164523 10.11113/mjfas.v18n4.2580 2-s2.0-85143418044 4 18 489 496 en Malaysian Journal of Fundamental and Applied Sciences © 2022 Kai Siong Yow, Siti Hasana Sapar, Cheng Yaw Low. This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited. application/pdf |
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Engineering::Computer science and engineering Diophantine Equation Integral Solution Yow, Kai Siong Sapar, Siti Hasana Low, Cheng Yaw Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ |
| description |
We present a method of determining integral solutions to the equation x2 + 16 ∙ 7b = y2r, where x, y, b, r ∈ ℤ+. We observe that the results can be classified into several categories. Under each category, a general formula is obtained using the geometric progression method. We then provide the bound for the number of non-negative integral solutions associated with each b. Lastly, the general formula for each of the categories is obtained and presented to determine the respective values of x and yr. We also highlight two special cases where different formulae are needed to represent their integral solutions. |
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School of Computer Science and Engineering |
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School of Computer Science and Engineering Yow, Kai Siong Sapar, Siti Hasana Low, Cheng Yaw |
| format |
Article |
| author |
Yow, Kai Siong Sapar, Siti Hasana Low, Cheng Yaw |
| author_sort |
Yow, Kai Siong |
| title |
Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ |
| title_short |
Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ |
| title_full |
Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ |
| title_fullStr |
Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ |
| title_full_unstemmed |
Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ |
| title_sort |
solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ |
| publishDate |
2023 |
| url |
https://hdl.handle.net/10356/164523 |
| _version_ |
1757048199238909952 |