On Euler's magic matrices of sizes $3$ and $8$

Abstract

A proper Euler's magic matrix is an integer $n\times n$ matrix $M\in {\mathbb{Z}}^{n\times n}$ such that $M\cdot M^t={\hat{I}}^3\cdot I$ for some nonzero constant ${\hat{I}}^3$, the sum of the squares of the entries along each of the two main diagonals equals ${\hat{I}}^3$, and the squares of all entries in $M$ are pairwise distinct. Euler constructed such matrices for $n=4$. In this work, we construct examples for $n=8$ and prove that no such matrix exists for $n=3$.

13 pages; enhanced arguments; some examples concerning the case $n=5$; to be published in Acta Arithmetica