AI 中文总结
研究5阶恰当欧拉幻方矩阵的构造问题，核心方法是在镜面对称坐标对下旋转Müller的“近似解”，使对角条件归结为有理方程，主要贡献是成功构造出5阶恰当欧拉幻方矩阵并为奇数阶提供统一方法。
AI 中文摘要
欧拉幻方矩阵是一个整数矩阵\(M\)，满足\(MM^{t}=\gamma I\)，其主对角线上元素平方和为\(\gamma\)；若其平方元素两两不同，则为恰当的。欧拉构造了一个4阶恰当例子，Müller解决了3阶（不存在）和8阶的情况，5阶是最小的未解决情况。我们通过在镜面对称坐标对下旋转Müller的一个“近似解”来构造这样一个矩阵，使两个对角条件归结为一个有理方程；相同的不变量为奇数阶提供了统一方法。
英文摘要
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.
Comments10 pages, plus verification source code
