AI 中文总结
本文构造了6×6伪双幻方和8×8双幻方的参数化族，并推广到6m阶和8m阶，给出数值示例。
AI 中文摘要
一个 $n$ 阶幻方是一个由 $n^2$ 个不同整数组成的方阵，使得每一行、每一列以及两条对角线上的整数之和都等于同一个公共和。如果一个幻方在将每个整数替换为其平方后仍然是幻方，则称之为双幻方。然而，如果上述平方方阵的所有行和所有列之和等于一个公共和，但两条对角线之和不等，则原始幻方被称为伪双幻方。除了一个 $6 \ imes 6$ 伪双幻方的族之外，参数化的双幻方族尚未被发表。在本文中，我们构造了 $6 \ imes 6$ 伪双幻方和 $8 \ imes 8$ 双幻方的参数化族。我们利用这些族，对任意正整数 $m$，构造了 $6m$ 阶伪双幻方和 $8m$ 阶双幻方的多参数族。我们还给出了 $6 \ imes 6$ 和 $12 \ imes 12$ 伪双幻方以及一个 $8 \ imes 8$ 双幻方的数值示例。
英文摘要
A magic square of order $n$ is a square array of $n^2$ distinct integers such that the integers of every row, every column and the two diagonals have the same common sum. If a magic square remains a magic square even when each integer is replaced by its square, it is called bimagic. If, however, all the rows and all the columns of the aforementioned square of squares add up to a common sum, but the two diagonals do not, the original magic square is called pseudo-bimagic. Except for one family of $6 \times 6$ pseudo-bimagic squares, parametrized families of bimagic squares have not been published. In this paper we construct parametrized families of $6 \times 6$ pseudo-bimagic squares and $8 \times 8$ bimagic squares. We use these families to construct, for any positive integer $m$, multi-parameter families of pseudo-bimagic squares of order $6m$ and bimagic squares of order $8m$. We also give numerical examples of $6 \times 6$ and $12 \times 12$ pseudo-bimagic squares and an $8 \times 8$ bimagic square.
Comments15 pages
