TITLE:
There Are Infinitely Many Mersnne Composite Numbers with Prime Exponents
AUTHORS:
Fengsui Liu
KEYWORDS:
Mersenne Composite Numbers, Sophie German Primes, Recursive Algorithm, Order Topology, Limit of Sequence of Sets
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.7,
July
31,
2018
ABSTRACT: 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 Mp.
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 Mp.