The classical Mersenne problem has been a stimulating challenge to number theorists and computer scientists for many years.
After briefly reviewing some of the natural settings in which this problem appears as a special case, we introduce an analogue
of the Mersenne problem in higher rank, in both a classical and an elliptic setting. Numerical evidence is presented for both
cases, and some of the difficulties involved in developing even a heuristic understanding of the problem are discussed.