EXISTENCE OF <i>K</i> -MULTIMAGIC SQUARES AND MAGIC SQUARES OF <i>k</i> th POWERS WITH DISTINCT ENTRIES

Abstract We demonstrate the existence of K -multimagic squares of order N consisting of $N^2$ distinct integers whenever $N&gt; 2K(K+1)$ . 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 k th powers when $$ \begin{align*}N&gt; \begin{cases} 2^{k+1} &amp; \text{if}\ 2 \leqslant k \leqslant 4, \\ 2 \lceil k(\log k + 4.20032) \rceil &amp; \text{if}\ k \geqslant 5, \end{cases}\end{align*} $$ improving on a recent result by Rome and Yamagishi [‘On the existence of magic squares of powers’, preprint (2024), arxiv:2406.09364].