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 n2 consecutive integers. Let A*d denote 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*e is an MS(n) for 1 ≤ e ≤ d. In this paper we investigate the existence of normal bimagic squares of order 2u and prove that there exists a normal bimagic square of order 2u, where u and 6 are coprime and u ≥ 5.