The present work aims at a small summary of Mersenne's prime numbers and, consequently, prime numbers, a subject that has shown to have a certain relevance because of the increase in users of applications that use RSA encryption to protect their data, being that it uses prime numbers in its calculations, especially primes with a large digit number, which is the case with Mersenne primes. We will approach themes that are crucial in Number Theory for the understanding of this complex subject, on which we present some properties and definitions, recent advances in the case of conjectures, and the use of the Lucas-Lehmer test to know if a number is prime or composite. We will cite the biography of Marin Mersenne and particular cases of primes that have been studied, the case of perfect numbers.
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