Mathematics Colloquia and Seminars

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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 $2 \le k \le K$. Given $K \ge 2$, we consider the problem of establishing the smallest integer $N(K)$ for which there exists non-trivial $K$-multimagic squares of order $N(K)$. Previous results on multimagic squares show that $N(K) \le (4K-2)^K$ for large $K$. Here we utilize the Hardy-Littlewood circle method and establish the bound \[N_2(K) \le 2K(K+1)+1.\] We additionally address the simpler problem of magic squares consisting of $k$th powers, improving on a recent result by Rome and Yamagishi.