Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

A proper Euler’s magic matrix is an integer $n\times n$ matrix $M\in \mathbb Z^{n\times n}$ such that $M\cdot M^t=\gamma \cdot I$ 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 $n=3$.

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