# Permutations of Maximal Order

Landau considered the following question – What is the maximal order of an element in the permutation group on k letters? We’ll prove some elementary bounds and deduce an asymptotic using the PNT.

Does there exist a pair of loaded six-sided dice such that the probability of rolling any dice sum {2,..12} is equally likely? We’ll show how this and other related questions about dice sums can be analyzed using cyclotomic polynomials.