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

Technical Summary: On Euler's Magic Matrices of Sizes 3 and 8

Problem Statement
The paper investigates "Euler's magic matrices," a specific class of integer matrices $M \in \mathbb{Z}^{n \times n}$ defined by three conditions for some nonzero constant $\gamma$:

1. Orthogonality-like condition: $M \cdot M^t = \gamma \cdot I_n$.
2. Main diagonal condition: The sum of the squares of the entries on the main diagonal equals $\gamma$.
3. Anti-diagonal condition: The sum of the squares of the entries on the anti-diagonal equals $\gamma$.

If the squares of the entries in $M$ are pairwise distinct, the matrix is termed a "proper" Euler's magic matrix. The existence of such matrices relates to the construction of "magic squares of squares" (matrices where entries are distinct squares and row/column/diagonal sums are equal). While Euler constructed such matrices for $n=4$, the existence for $n=3$ remains an open problem in the context of magic squares of squares. This paper addresses the existence of Euler's magic matrices specifically for $n=3$ and $n=8$, and provides constructions for other $n$.

Methodology

• Case $n=3$: The author employs a reduction to orthogonal matrices over the field of rational numbers ($\mathbb{Q}$). Using Lemma 2.1, it is established that for odd $n$, any Euler's magic matrix $M$ with $M \cdot M^t = \gamma I$ can be scaled to be orthogonal. The proof utilizes a refinement of the Cayley transform (Proposition 2.2), parametrizing orthogonal matrices via skew-symmetric matrices $S$. By substituting this parametrization into the diagonal and anti-diagonal sum conditions, the problem reduces to solving a system of polynomial equations in the entries of $S$. The author demonstrates that these equations lead to a contradiction over $\mathbb{Q}$ (specifically involving the non-square nature of 3 in $\mathbb{Q}$), proving non-existence.

• Case $n=8$: The construction relies on the algebra of octonions. The author defines $L(a, \dots, h)$ and $R(p, \dots, w)$ as matrices representing left and right multiplication by octonions, respectively. The product $M = L \cdot R$ automatically satisfies the orthogonality condition $M \cdot M^t = \gamma I_8$. The challenge is reduced to satisfying the diagonal and anti-diagonal conditions (equations 2 and 3) while ensuring the "properness" condition (distinct squares).

  • Search Strategy: The paper details a "naive search" approach which proved difficult due to the complexity of the resulting quartic curves and the difficulty of finding rational points.
  • Algebraic Reduction: A more successful strategy involved imposing specific algebraic restrictions on the parameters of $L$ (specifically $h = \pm a$ and a sum-of-squares condition on the remaining parameters). This reduced the degree of the polynomial system, allowing for the elimination of variables and the derivation of a rational parametrization.

• General $n \geq 4$: The paper constructs "improper" Euler's magic matrices (where entries are not necessarily distinct squares) using permutation matrices derived from specific cycle structures in the symmetric group.

Key Contributions and Results

1. Non-existence for $n=3$: The paper proves Theorem 1.2, stating that no Euler's magic matrix exists in $\mathbb{Q}^{3 \times 3}$. Consequently, no proper Euler's magic matrix exists for $n=3$. The proof relies on showing that the system of equations derived from the Cayley transform has no rational solutions.

2. Existence for $n=8$: The paper provides the first explicit construction of a proper Euler's magic matrix for $n=8$.

   • Theorem 1.3 presents a specific $8 \times 8$ integer matrix $M = L \cdot R$ with distinct squared entries, constructed using specific integer parameters for the octonion multiplication matrices.
   • Theorem 1.4 generalizes this result, providing a 4-parametric family of proper Euler's magic matrices over $\mathbb{Q}(q, r, t, u)$. This demonstrates that solutions are not isolated but part of a larger algebraic family.

3. Constructions for other $n$:

   • Theorem 4.1 constructs Euler's magic matrices for all $n \geq 4$ (both even and odd) using permutation matrices. However, the author explicitly notes these are "far from proper" as they contain many zero entries and repeated values.
   • For $n=5$, the paper reports on a random search that yielded matrices satisfying the Euler conditions but failing the "proper" condition (distinct squares) by a small margin (e.g., 24 distinct squares out of 25 entries). The existence of a proper matrix for $n=5$ remains an open question.

Significance and Claims

The paper claims to resolve the existence question for Euler's magic matrices in the specific cases of $n=3$ and $n=8$.

• For $n=3$, it definitively closes the door on the existence of such matrices over the rationals, suggesting that the difficulty in finding a $3 \times 3$ magic square of squares may be deeper than previously suspected, as even the relaxed "Euler's magic matrix" condition cannot be met.
• For $n=8$, it validates a suggestion by Isabel Pirsic to use octonion multiplication matrices. While Pirsic's massive search failed to find examples, this paper demonstrates that a careful algebraic analysis of the polynomial system yields solutions without the need for exhaustive searching.
• The work highlights that $n=3$ is the unique positive integer for which no Euler's magic matrix exists in $\mathbb{Q}^{n \times n}$, while such matrices exist for all other $n$ (though properness is only guaranteed for $n=4$ and $n=8$ in this work).

The author maintains a modest tone regarding the $n=5$ case, acknowledging that while near-proper examples were found, the existence of a fully proper matrix remains unproven. The paper emphasizes the utility of algebraic structures (quaternions for $n=4$, octonions for $n=8$) in generating these matrices and the power of polynomial system analysis over brute-force search.