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

We demonstrate the existence of $K$-multimagic squares of order $N$ consisting of distinct integers whenever $N>2K(K+1)$. 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 $$N>\{\begin{array}{ll}{2}^{k+1}& \text{ if }2⩽k⩽4\\ 2\lceil k(\log k+4.20032)\rceil & \text{ if }k⩾5\end{array}$$ improving on a recent result by Rome and Yamagishi.