On Euler’s magic matrices of sizes 3 and 8

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

Published in Open Access (under CC-BY license).