A proper Euler magic matrix of order $5$

Abstract

An Euler magic matrix is an integer matrix $M$ with $MM^{t}=γI$ whose squared entries sum to $γ$ along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler constructed an order-$4$ proper example, and Müller settled orders $3$ (none exist) and $8$, leaving order $5$ as the smallest open case. We construct such a matrix, by rotating one of Müller's "near-misses" under a mirror-symmetric coordinate pair so that the two diagonal conditions collapse to a single rational equation; the same invariant suggests a uniform approach to the odd orders.

Disclosure

“∗ Harvard Business School; Department of Economics and Center of Mathematical Sciences and Applications, Harvard University; and a16z crypto. I used LLMs to assist with computations and coding in the preparation of this article, especially Claude Opus 4.8 and GPT-5.5 Pro (both accessed in part via Poe with the”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T18:57:54.558420Z

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