Existence of $K$-multimagic squares and magic squares of $k$th powers with distinct entries
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929653371895808 |
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| author | Flores, Daniel |
| author_facet | Flores, Daniel |
| contents | We demonstrate the existence of $K$-multimagic squares of order $N$ consisting of distinct integers whenever $N>2 K(K+1)$. This improves upon our earlier result in which we only required $N+1$ distinct integers.
Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct $k$ th powers when
$$ N> \begin{cases}2^{k+1} & \text { if } 2 \leqslant k \leqslant 4 \\ 2\lceil k(\log k+4.20032)\rceil & \text { if } k \geqslant 5\end{cases} $$
improving on a recent result by Rome and Yamagishi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01091 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of $K$-multimagic squares and magic squares of $k$th powers with distinct entries Flores, Daniel Number Theory Combinatorics 11D45, 11D72, 11P55, 11E76, 11L07, 05B15, 05B20 We demonstrate the existence of $K$-multimagic squares of order $N$ consisting of distinct integers whenever $N>2 K(K+1)$. This improves upon our earlier result in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct $k$ th powers when $$ N> \begin{cases}2^{k+1} & \text { if } 2 \leqslant k \leqslant 4 \\ 2\lceil k(\log k+4.20032)\rceil & \text { if } k \geqslant 5\end{cases} $$ improving on a recent result by Rome and Yamagishi. |
| title | Existence of $K$-multimagic squares and magic squares of $k$th powers with distinct entries |
| topic | Number Theory Combinatorics 11D45, 11D72, 11P55, 11E76, 11L07, 05B15, 05B20 |
| url | https://arxiv.org/abs/2411.01091 |