Existence of $K$-multimagic squares and magic squares of $k$th powers with distinct entries

TL;DR

This paper proves the existence of large $K$-multimagic squares with distinct entries and provides a method to construct magic squares of $k$th powers, improving previous bounds on their size.

Contribution

It establishes new bounds for the existence of $K$-multimagic squares with distinct entries and introduces a direct construction method for magic squares of $k$th powers.

Findings

Existence of $K$-multimagic squares for $N > 2K(K+1)$

Existence of magic squares of $k$th powers for specific bounds on $N$

Improvement over previous results by Rome and Yamagishi

Abstract

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.