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Abstract

Doing research on a number has become a study project for a lot of mathematicians since the start of the Pythagoras School in Greece in 300 BC. Especially, since Euclid proved the fundamental theorem of arithmetic, a prime has been playing a great role in the research on the number theory. Amongst all, a lot of mathematicians have conducted the research work on the numbers of having a special form, and Mersenne's prime belongs to one of them. The Mersenne's number having the form like when is an integer is called Mersenne prime when is a prime( is a prime). Such Mersenne number is called so by adding the name of the 17th century mathematician Marin Mersenne, and due to its special form, it is used for modern cryptology and performance test of the computer; in addition, the Mersenne number is also the biggest form of a prime ever known up to the present. Accordingly, in Chapter II of this thesis, this research is going to delve into the Mersenne number more deeply after doing explanation about the contents of the basic number theory needed for the understanding of the properties of a number. In Chapter III, this research intends to look at the judging method for finding the Mersenne numbers after looking at the definition of the Mersenne number through Mersenne's life and also looking at the process of finding the Mersenne prime together with the general properties of Mersenne's numbers. In addition, this research introduces the unsolved problems with the Mersenne numbers.