A circle method approach to K-multimagic squares
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912173948665856 |
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| author | Flores, Daniel |
| author_facet | Flores, Daniel |
| contents | In this paper we investigate $K$-multimagic squares of order $N$, these are $N \times N$ magic squares which remain magic after raising each element to the $k$ th power for all $2 \leqslant$ $k \leqslant K$. Given $K \geqslant 2$, we consider the problem of establishing the smallest integer $N_2(K)$ for which there exists nontrivial $K$-multimagic squares of order $N_2(K)$. Previous results on multimagic squares show that $N_2(K) \leqslant(4 K-2)^K$ for large $K$. Here we utilize the Hardy-Littlewood circle method and establish the bound
$$ N_2(K) \leqslant 2 K(K+1)+1 $$
Via an argument of Granville's we additionally deduce the existence of infinitely many nontrivial prime valued $K$-multimagic squares of order $2 K(K+1)+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08161 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A circle method approach to K-multimagic squares Flores, Daniel Number Theory Combinatorics 11D45, 11D72, 11P55, 11E76, 11L07, 05B15, 05B20 In this paper we investigate $K$-multimagic squares of order $N$, these are $N \times N$ magic squares which remain magic after raising each element to the $k$ th power for all $2 \leqslant$ $k \leqslant K$. Given $K \geqslant 2$, we consider the problem of establishing the smallest integer $N_2(K)$ for which there exists nontrivial $K$-multimagic squares of order $N_2(K)$. Previous results on multimagic squares show that $N_2(K) \leqslant(4 K-2)^K$ for large $K$. Here we utilize the Hardy-Littlewood circle method and establish the bound $$ N_2(K) \leqslant 2 K(K+1)+1 $$ Via an argument of Granville's we additionally deduce the existence of infinitely many nontrivial prime valued $K$-multimagic squares of order $2 K(K+1)+1$. |
| title | A circle method approach to K-multimagic squares |
| topic | Number Theory Combinatorics 11D45, 11D72, 11P55, 11E76, 11L07, 05B15, 05B20 |
| url | https://arxiv.org/abs/2406.08161 |