On a binary Diophantine inequality involving prime numbers
Let $ N $ denote a sufficiently large real number. In this paper, we prove that for $ 1 < c < \frac{104349}{77419} $, $ c\neq\frac{4}{3} $, for almost all real numbers $ T\in(N, 2N] $ (in the sense of Lebesgue measure), the Diophantine inequality $ |p_1^c+p_2^c-T| < T^{-\frac{9}...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-02-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2024407?viewType=HTML |
| _version_ | 1797266474927128576 |
|---|---|
| author | Jing Huang Qian Wang Rui Zhang |
| author_facet | Jing Huang Qian Wang Rui Zhang |
| author_sort | Jing Huang |
| collection | DOAJ |
| description | Let $ N $ denote a sufficiently large real number. In this paper, we prove that for $ 1 < c < \frac{104349}{77419} $, $ c\neq\frac{4}{3} $, for almost all real numbers $ T\in(N, 2N] $ (in the sense of Lebesgue measure), the Diophantine inequality $ |p_1^c+p_2^c-T| < T^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)} $ is solvable in primes $ p_1, p_2 $. In addition, it is proved that the Diophantine inequality $ |p_1^c+p_2^c+p_3^c+p_4^c-N| < N^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)} $ is solvable in primes $ p_1, p_2, p_3, p_4 $. This result constitutes a refinement upon that of Li and Cai. |
| first_indexed | 2024-04-25T01:01:16Z |
| format | Article |
| id | doaj.art-12c613f8cd144ab298cc39f87c15a341 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-25T01:01:16Z |
| publishDate | 2024-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-12c613f8cd144ab298cc39f87c15a3412024-03-11T01:40:03ZengAIMS PressAIMS Mathematics2473-69882024-02-01948371838510.3934/math.2024407On a binary Diophantine inequality involving prime numbersJing Huang0Qian Wang1Rui Zhang2School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan 250014, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan 250014, ChinaLet $ N $ denote a sufficiently large real number. In this paper, we prove that for $ 1 < c < \frac{104349}{77419} $, $ c\neq\frac{4}{3} $, for almost all real numbers $ T\in(N, 2N] $ (in the sense of Lebesgue measure), the Diophantine inequality $ |p_1^c+p_2^c-T| < T^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)} $ is solvable in primes $ p_1, p_2 $. In addition, it is proved that the Diophantine inequality $ |p_1^c+p_2^c+p_3^c+p_4^c-N| < N^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)} $ is solvable in primes $ p_1, p_2, p_3, p_4 $. This result constitutes a refinement upon that of Li and Cai.https://www.aimspress.com/article/doi/10.3934/math.2024407?viewType=HTMLdiophantine inequalityprimeexponential sum |
| spellingShingle | Jing Huang Qian Wang Rui Zhang On a binary Diophantine inequality involving prime numbers AIMS Mathematics diophantine inequality prime exponential sum |
| title | On a binary Diophantine inequality involving prime numbers |
| title_full | On a binary Diophantine inequality involving prime numbers |
| title_fullStr | On a binary Diophantine inequality involving prime numbers |
| title_full_unstemmed | On a binary Diophantine inequality involving prime numbers |
| title_short | On a binary Diophantine inequality involving prime numbers |
| title_sort | on a binary diophantine inequality involving prime numbers |
| topic | diophantine inequality prime exponential sum |
| url | https://www.aimspress.com/article/doi/10.3934/math.2024407?viewType=HTML |
| work_keys_str_mv | AT jinghuang onabinarydiophantineinequalityinvolvingprimenumbers AT qianwang onabinarydiophantineinequalityinvolvingprimenumbers AT ruizhang onabinarydiophantineinequalityinvolvingprimenumbers |