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Abstract. 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. 1.

We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65-digit (2^3539+ l)/3, the first ordinary Titanic prime.

In this note, we give a necessary condition for the primality of (2 p + 1)/3. 1