三次幻方的构作

给出2k维m阶t次幻方及m模方阵,m模列满秩矩阵,模线,m经典模线集和t次m模基因阵的概念,并用矩阵法和组合法初步研究了t次幻方特别是三次幻方的构作.证明:(i)若存在2k阶t次m模基因阵,则存在2k维m阶t次幻方;(ii)若N=P1α1P2α2…PSαS为N的标准分解式,iα≥3,Piiα≥16(1≤i≤S),则存在二维N阶三次幻方;(iii)若存在二维偶m阶2t+1次幻方和二维n阶2t次幻方,则存在二维mn阶2t+1次幻方;(iv)若存在二维m阶和n阶t次幻方,则存在二维mn阶t次幻方;(v)当t≥3时,不存在二维单偶数阶t次幻方.

On the Construction of Trimagic Squares

We presented the following concepts: t-multimagic square,m-modulo matrix,m-modulo column linear independent matrix,modulo line,and m-modulo gene matrix.Using the gene matrices and the combination methods,we put forward simple constructing methods of t-multimagic squares,especially trimagic squares.We proved that(i) there exists a t-multimagic square of dimension 2k of order m if there exists a t-multimagic m-modulo gene matrix.(ii) there exists a trimagic square of order N if N=Pα11Pα22…PαSS where Pi is prime,αi≥3 and Pαii≥16.(iii) if there is a(2t+1)-multimagic square of order m where m is even and there is a 2t-multimagic square of order n,then there exists a(2t+1)-multimagic square of order mn.(iv) if there are the t-multimagic squares of order m and n,then there exists a t-multimagic square of order mn.(v) there does not exist a t-multimagic square of order m if m=2k,k is odd,and t≥3.