6阶正则欧拉幻矩阵
A proper Euler magic matrix of order 6
浏览论文内容
中文总结 AI 辅助
本文针对未被前人解决的6阶正则欧拉幻矩阵问题，给出了γ=18500和γ=43290的两个构造，并确定其计数下界为γ≥2485。
中文摘要 AI 辅助
欧拉幻矩阵是一种整数矩阵M，满足MMᵀ=γI（其中γ≠0），且其平方元素在两条主对角线上的和均为γ；若其平方元素两两不同，则称其为正则欧拉幻矩阵。欧拉给出了1个4阶正则欧拉幻矩阵的例子，Müller确定了3阶不存在、8阶存在，Kominers确定了5阶存在。我们针对Müller和Kominers未解决的6阶情况，给出了1个γ=18500的6阶正则欧拉幻矩阵的构造，以及另1个独立的γ=43290的6阶正则欧拉幻矩阵；证明方式为显式矩阵构造与有限精确验证。我们还记录了6阶正则欧拉幻矩阵的基本计数下界γ≥2485。
英文摘要
An Euler magic matrix is an integer matrix M with MM^t = gamma I for some gamma != 0, whose squared entries sum to gamma along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler gave a proper example of order 4; Müller settled orders 3 (none exists) and 8; and Kominers settled order 5. We give an order-6 construction, a case not addressed by Müller or Kominers, exhibiting a proper Euler magic matrix of order 6 with gamma = 18500 together with a second, independent one with gamma = 43290. The proof is the explicit matrix and a finite exact verification. We also record an elementary counting bound gamma >= 2485 for proper order-6 examples.
