用范德蒙行列式构造协调正交对角线拉丁D体和双重标准幻D体

Construction of Harmonious Orthogonal Diagonal Latin D-cubes and Addition-Multiplication Perfect Magic D-cubes Using Vandermonde determinant

要所谓ｎ阶双重标准幻Ｄ体是一个ｎ阶Ｄ维阵列，它的元是ｎ￣Ｄ个不同的正整数，使得每一“行”，每一“对角线”上的－ｎ个元不仅和，而且积均为常数．在本文中，我们给出了用范德蒙行列式构造协调正交对角线拉丁Ｄ体和用协调正交对角线拉丁Ｄ体构造双重标准幻Ｄ体的方法，从而证明了：对于所有２≤Ｄ≤ｋ，存在常数Ｃ＿（ｋ，Ｄ），当ｍ的最小素因子大于Ｃ＿（ｋ，Ｄ）时，至少存在一个ｍ￣ｋ阶标准幻Ｄ体．特别地，Ｃ＿（ｋ，２）＝２；当ｋ≥３时，Ｃ＿（ｋ，２）＝１；Ｃ＿（ｋ，３）≤７．

n addition-multiplication perfect magic D-cube of order n is an n×n ×…×n D-dimensionalarray whose elements are n￣D distinct positive integers such that not only the sum but also theproduct of the elements in each“row”,“diagonal”is a constant.In this paper,we give a construc-tion of harmonious orthogonal latin D-cubes and addition-multiplication perfect magic D-cubesusing vandermonde deter minant,and show that for every k≥D≥2,there a constant C_(k,D) such thatthere is an addition- multiplication perfect magic D- cube of order m￣k if the minimal prime divisor ofan positive integer m is greater than C_(k,D),Especially,we show that C_(k,2)=2,C_(k,2)=1 if k≥3,C_(k,3)≤7.