دورية أكاديمية
An application of p-adic Baker method to a special case of Jeśmanowicz' conjecture
| العنوان: | An application of p-adic Baker method to a special case of Jeśmanowicz' conjecture |
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| المؤلفون: | Ziyu Dong, Zhengjun Zhao |
| المصدر: | AIMS Mathematics, Vol 8, Iss 5, Pp 11617-11628 (2023) |
| بيانات النشر: | AIMS Press, 2023. |
| سنة النشر: | 2023 |
| المجموعة: | LCC:Mathematics |
| مصطلحات موضوعية: | exponential diophantine equation, pythagorean triple, jeśmanowicz' conjecture, Mathematics, QA1-939 |
| الوصف: | In 1956, Jeśmanowicz conjectured that, for any positive integer $ n $, the Diophantine equation $ \left((f^{2}-g^{2})n\right)^{x}+\left((2fg)n\right)^{y} = \left((f^{2}+g^{2})n\right)^z $ has only the positive integral solution $ (x, y, z) = (2, 2, 2) $, where $ f $ and $ g $ are positive integers with $ f > g $, gcd$ (f, g) = 1 $, and $ f\not\equiv g\pmod {2} $. Let $ r = 6k+2 $, $ k \in \mathbb{N} $, $ k\geq25 $. In this paper, combining $ p $-adic form of Baker method with some detailed computation, we prove that if $ n $ satisfies $ n\equiv 0, 6, 9\pmod{12} $, $ f = g+1 $ and $ g = 2^{r}-1 $, then the conjecture is true. |
| نوع الوثيقة: | article |
| وصف الملف: | electronic resource |
| اللغة: | English |
| تدمد: | 2473-6988 |
| Relation: | https://doaj.org/toc/2473-6988 |
| DOI: | 10.3934/math.2023588?viewType=HTML |
| DOI: | 10.3934/math.2023588 |
| URL الوصول: | https://doaj.org/article/aa339b57298848a69ab77c0a572b36e3 |
| رقم الأكسشن: | edsdoj.339b57298848a69ab77c0a572b36e3 |
| قاعدة البيانات: | Directory of Open Access Journals |
Full Text Finder | تدمد: | 24736988 |
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| DOI: | 10.3934/math.2023588?viewType=HTML |