Expression for $g(k)$ Related to Waring's Problem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918135568793600 |
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| author | Root, Owen |
| author_facet | Root, Owen |
| contents | Waring's Problem asks whether, for each positive integer $k$, there exists an integer $s$ such that every positive integer is a sum of at most $k$th powers. While Hilbert proved the existence of such $s$, Waring's Problem has lead to areas of related work, namely the function $g(k)$, which denotes the least such $s$. There is no known general closed form for $g(k)$, though for $g(k)$ has been evaluated for small $k$. Prior work has reduced the problem to verifying a particular condition, which if never occurs, implies an expression for $g(k)$. In this paper, I present a proof the condition never occurs, thus fixing the value of $g(k)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17950 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Expression for $g(k)$ Related to Waring's Problem Root, Owen Number Theory Waring's Problem asks whether, for each positive integer $k$, there exists an integer $s$ such that every positive integer is a sum of at most $k$th powers. While Hilbert proved the existence of such $s$, Waring's Problem has lead to areas of related work, namely the function $g(k)$, which denotes the least such $s$. There is no known general closed form for $g(k)$, though for $g(k)$ has been evaluated for small $k$. Prior work has reduced the problem to verifying a particular condition, which if never occurs, implies an expression for $g(k)$. In this paper, I present a proof the condition never occurs, thus fixing the value of $g(k)$. |
| title | Expression for $g(k)$ Related to Waring's Problem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.17950 |