The sufficient conditions for insolvability of some Diophantine equations of $n$-th degree
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908361711157248 |
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| author | Samsonadze, Eteri |
| author_facet | Samsonadze, Eteri |
| contents | The sufficient conditions for insolvability of the Diophantine equation $\sum_{i=1}^{m}x_i^{n}=bc^{n}$ ($n, m \geq 2$, $b, c\in \mathbb{N}$) in nonnegative integers are obtained for the case where the canonical decomposition of the number $c$ consists of powers of primes $p_i$ which satisfy the condition $φ(p_i^{k_i})\mid n$ ($p_i^{k_i }\geq 3)$ for some natural numbers $k_i$ $(i=1,2,\ldots ,l)$; $φ(x)$ is the Euler's totient function. Moreover, it is proved that if $b< m< p_i^{k_i}$ $(i=1,2,\ldots ,l)$, then this equation has no solution with natural components $x_1,x_2,\ldots ,x_m$. Besides, applying only elementary methods, it is proved that the Diophantine equation $x_1^n+x_2^n=(p^{s} p_1^{s_1} p_2^{s_2}\ldots p_l^{s_l})^{n}$ (with nonnegative integers $s$, $s_i$ $(i=1,2,..,l)$) has no solution with natural components if $n\geq 3$, $p$ is a prime number, while $p_i$ is a prime such that there is a natural number $k_i$ with $φ(p_i^{k_i})\mid n$ $(p_i^{k_i}\geq 3)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_07878 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The sufficient conditions for insolvability of some Diophantine equations of $n$-th degree Samsonadze, Eteri Number Theory 11D41, 11D72 The sufficient conditions for insolvability of the Diophantine equation $\sum_{i=1}^{m}x_i^{n}=bc^{n}$ ($n, m \geq 2$, $b, c\in \mathbb{N}$) in nonnegative integers are obtained for the case where the canonical decomposition of the number $c$ consists of powers of primes $p_i$ which satisfy the condition $φ(p_i^{k_i})\mid n$ ($p_i^{k_i }\geq 3)$ for some natural numbers $k_i$ $(i=1,2,\ldots ,l)$; $φ(x)$ is the Euler's totient function. Moreover, it is proved that if $b< m< p_i^{k_i}$ $(i=1,2,\ldots ,l)$, then this equation has no solution with natural components $x_1,x_2,\ldots ,x_m$. Besides, applying only elementary methods, it is proved that the Diophantine equation $x_1^n+x_2^n=(p^{s} p_1^{s_1} p_2^{s_2}\ldots p_l^{s_l})^{n}$ (with nonnegative integers $s$, $s_i$ $(i=1,2,..,l)$) has no solution with natural components if $n\geq 3$, $p$ is a prime number, while $p_i$ is a prime such that there is a natural number $k_i$ with $φ(p_i^{k_i})\mid n$ $(p_i^{k_i}\geq 3)$. |
| title | The sufficient conditions for insolvability of some Diophantine equations of $n$-th degree |
| topic | Number Theory 11D41, 11D72 |
| url | https://arxiv.org/abs/2505.07878 |