1 results
EXISTENCE OF K-MULTIMAGIC SQUARES AND MAGIC SQUARES OF kth POWERS WITH DISTINCT ENTRIES
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 11 February 2025, pp. 1-8
-
- Article
- Export citation
-
We demonstrate the existence of K-multimagic squares of order N consisting of
$N^2$ distinct integers whenever 
$N> 2K(K+1)$. This improves our earlier result [D. Flores, ‘A circle method approach to K-multimagic squares’, preprint (2024), arXiv:2406.08161] 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 kth powers when 
$$ \begin{align*}N> \begin{cases} 2^{k+1} & \text{if}\ 2 \leqslant k \leqslant 4, \\ 2 \lceil k(\log k + 4.20032) \rceil & \text{if}\ k \geqslant 5, \end{cases}\end{align*} $$improving on a recent result by Rome and Yamagishi [‘On the existence of magic squares of powers’, preprint (2024), arxiv:2406.09364].
