On solutions of $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910841512656896 |
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| author | Louwsma, Joel |
| author_facet | Louwsma, Joel |
| contents | We give an algorithm that produces all solutions of the equation $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$, where $k$ is a fixed positive integer that is not a power of $2$, $a$ is an element of $\{0,1,2\}$ that can vary from term to term, and $b$ is a nonnegative integer that can vary from term to term. We also completely characterize the pairs $(k,n)$ for which this equation has a nontrivial solution in integers of this form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09515 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On solutions of $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$ Louwsma, Joel Number Theory Combinatorics 11D68 (Primary) 11D45, 11D72, 11Y50 (Secondary) We give an algorithm that produces all solutions of the equation $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$, where $k$ is a fixed positive integer that is not a power of $2$, $a$ is an element of $\{0,1,2\}$ that can vary from term to term, and $b$ is a nonnegative integer that can vary from term to term. We also completely characterize the pairs $(k,n)$ for which this equation has a nontrivial solution in integers of this form. |
| title | On solutions of $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$ |
| topic | Number Theory Combinatorics 11D68 (Primary) 11D45, 11D72, 11Y50 (Secondary) |
| url | https://arxiv.org/abs/2402.09515 |