A Construction of Addition-Multiplication Magic Squares Using Orthogonal Diagonal Latin Squares

An addition-multiplication magic square of order \(n\) is an \(n \times n\) matrix whose entries are \(n^2\) distinct positive integers such that not only the sum but also the product of the entries in each row, column, main diagonal, and back diagonal is a constant. It is shown in this paper that such a square exists for any order \(mn\), where \(m\) and \(n\) are positive integers and \(m, n \notin \{1, 2, 3, 6\}\).