Integers that are not the sum of positive powers
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915218970378240 |
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| author | Benfield, Brennan Lippard, Oliver |
| author_facet | Benfield, Brennan Lippard, Oliver |
| contents | The generalized Waring problem asks exactly which positive integers cannot be expressed as the sum of $j$ positive $k$-th powers? Using computational techniques, this paper refines an approach introduced by Zenkin, establishes results for the individual cases $5 \le k \le 9$, and resolves conjectures of Zenkin and the OEIS. This paper further establishes theoretical results regarding the properties of the sets of integers that are not the sum of $j$ positive $k$-th powers. The notion of Waring's problem is further extended to the finite sets of non-representable numbers where $G(1,k) < j < g(1,k)$. Improved computational techniques and results from Waring's problem are used throughout to catalog the sets of such integers, which are then considered in a general setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08193 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integers that are not the sum of positive powers Benfield, Brennan Lippard, Oliver Number Theory 11P05, 11-04, 11P81, 11Y50 The generalized Waring problem asks exactly which positive integers cannot be expressed as the sum of $j$ positive $k$-th powers? Using computational techniques, this paper refines an approach introduced by Zenkin, establishes results for the individual cases $5 \le k \le 9$, and resolves conjectures of Zenkin and the OEIS. This paper further establishes theoretical results regarding the properties of the sets of integers that are not the sum of $j$ positive $k$-th powers. The notion of Waring's problem is further extended to the finite sets of non-representable numbers where $G(1,k) < j < g(1,k)$. Improved computational techniques and results from Waring's problem are used throughout to catalog the sets of such integers, which are then considered in a general setting. |
| title | Integers that are not the sum of positive powers |
| topic | Number Theory 11P05, 11-04, 11P81, 11Y50 |
| url | https://arxiv.org/abs/2404.08193 |