Magic \(N\)-cubes form a free monoid (Q1378504)

Summary: We prove a conjecture stated in an earlier paper [\textit{A. Adler} and \textit{S. R. Li}, Am. Math. Mon. 84, 618-627 (1977; Zbl 0389.05018)]. The conjecture states that with respect to a rather natural operation, the set of \(N\)-dimensional magic cubes forms a free monoid for every integer \(N>1\). A consequence of this conjecture is a certain identity of formal Dirichlet series. These series and the associated power series are shown to diverge. Generalizations of the underlying ideas are presented. We also prove variants of the main results for magic cubes with remarkable power sum properties.