Construction of Normal Bimagic Squares of Order 2u

An n × n matrix A consisting of nonnegative integers is a general magic square of order n if the sum of elements in each row, column, and main diagonal is the same. A general magic square A of order n is called a magic square, denoted by MS（n）, if the entries of A are distinct. A magic square A of order n is normal if the entries of A are n;consecutive integers. Let A*ddenote the matrix obtained by raising each element of A to the d-th power. The matrix A is a d-multimagic square, denoted by MS（n, d）, if A;is an MS（n） for 1 ≤ e ≤ d. In this paper we investigate the existence of normal bimagic squares of order 2 u and prove that there exists a normal bimagic square of order 2u, where u and 6 are coprime and u ≥ 5.