A note on consecutive integers of the form 2<sup><em>x</em></sup> + <em>y</em><sup><em>2</em></sup>
Let $k$ be a positive integer with $k\ge 2$. Let $N_k$ denote the number of $k$ tuples of consecutive integers with each of them in the form $2^x+y^2$, where $x,y$ are nonnegative integers. In this paper, we investigate the formulas for $N_k$. Actually, by using some elementary methods, we show that...
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-06-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/10.3934/math.2020285/fulltext.html |
| _version_ | 1828757095433895936 |
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| author | Zongbing Lin Kaimin Cheng |
| author_facet | Zongbing Lin Kaimin Cheng |
| author_sort | Zongbing Lin |
| collection | DOAJ |
| description | Let $k$ be a positive integer with $k\ge 2$. Let $N_k$ denote the number of $k$ tuples of consecutive integers with each of them in the form $2^x+y^2$, where $x,y$ are nonnegative integers. In this paper, we investigate the formulas for $N_k$. Actually, by using some elementary methods, we show that \[{N_k} = \left\{ \begin{array}{l} + \infty, \; \; \; \; \; {\rm{if}}\; {\rm{2}} \le k \le 4, \\ 6, \; \; \; \; \; \; \; \; \; {\rm{if}}\; k = 5, \\ 3, \; \; \; \; \; \; \; \; \; {\rm{if}}\; k = 6, \\ 0, \; \; \; \; \; \; \; \; \; {\rm{otherwise}}. \end{array} \right.\] |
| first_indexed | 2024-12-11T00:21:16Z |
| format | Article |
| id | doaj.art-0e06b32830be4e6bba8b8015dfce833b |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-11T00:21:16Z |
| publishDate | 2020-06-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-0e06b32830be4e6bba8b8015dfce833b2022-12-22T01:27:43ZengAIMS PressAIMS Mathematics2473-69882020-06-01554453445810.3934/math.2020285A note on consecutive integers of the form 2<sup><em>x</em></sup> + <em>y</em><sup><em>2</em></sup>Zongbing Lin0Kaimin Cheng11 School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, P. R. China2 School of Mathematics and Information, China West Normal University, Nanchong 637009, P. R. ChinaLet $k$ be a positive integer with $k\ge 2$. Let $N_k$ denote the number of $k$ tuples of consecutive integers with each of them in the form $2^x+y^2$, where $x,y$ are nonnegative integers. In this paper, we investigate the formulas for $N_k$. Actually, by using some elementary methods, we show that \[{N_k} = \left\{ \begin{array}{l} + \infty, \; \; \; \; \; {\rm{if}}\; {\rm{2}} \le k \le 4, \\ 6, \; \; \; \; \; \; \; \; \; {\rm{if}}\; k = 5, \\ 3, \; \; \; \; \; \; \; \; \; {\rm{if}}\; k = 6, \\ 0, \; \; \; \; \; \; \; \; \; {\rm{otherwise}}. \end{array} \right.\]https://www.aimspress.com/article/10.3934/math.2020285/fulltext.htmldiophantine equationconsecutive integerssum of two powers |
| spellingShingle | Zongbing Lin Kaimin Cheng A note on consecutive integers of the form 2<sup><em>x</em></sup> + <em>y</em><sup><em>2</em></sup> AIMS Mathematics diophantine equation consecutive integers sum of two powers |
| title | A note on consecutive integers of the form 2<sup><em>x</em></sup> + <em>y</em><sup><em>2</em></sup> |
| title_full | A note on consecutive integers of the form 2<sup><em>x</em></sup> + <em>y</em><sup><em>2</em></sup> |
| title_fullStr | A note on consecutive integers of the form 2<sup><em>x</em></sup> + <em>y</em><sup><em>2</em></sup> |
| title_full_unstemmed | A note on consecutive integers of the form 2<sup><em>x</em></sup> + <em>y</em><sup><em>2</em></sup> |
| title_short | A note on consecutive integers of the form 2<sup><em>x</em></sup> + <em>y</em><sup><em>2</em></sup> |
| title_sort | note on consecutive integers of the form 2 sup em x em sup em y em sup em 2 em sup |
| topic | diophantine equation consecutive integers sum of two powers |
| url | https://www.aimspress.com/article/10.3934/math.2020285/fulltext.html |
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