Author(s)

Fengsui Liu

Department of Mathematics, Nanchang University, Nanchang, China.

By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number M_p. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents M_p.

Mersenne Composite Numbers, Sophie German Primes, Recursive Algorithm, Order Topology, Limit of Sequence of Sets

Liu, F. (2018) There Are Infinitely Many Mersnne Composite Numbers with Prime Exponents. Advances in Pure Mathematics, 8, 687-698. doi: 10.4236/apm.2018.87041.