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 $M\cdot M^t=γ\cdot I$ for some nonzero constant $γ$, the sum of the squares of the entries along each of the two main diagonals equals $γ$, 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 $n=3$.