Mersenne Primes

 

정의1

1. A number of the form $2^{n} - 1$ is called a Mersenne number.

 

2. A Mersenne Number is called a Mersenne prime if it is a prime.

 

 

 

정리1

Let $a \geq 2$, $n \geq 2$  and  $a^{n} - 1$ is a prime.

$\Rightarrow$ $a=2$,  $n$ is a prime. 

 

  Since $a^{n} - 1$ is a prime and $ a^{n} - 1 = (a-1)(a^{n-1} + a^{n-2} + \cdots + 1)$,

  so $a-1 = 1$  $\Rightarrow$  $a = 2$

 

  Suppose 1 : $n = mk$  $(1 < m, k < n)$

  $ 2^{n} - 1 = 2^{mk} - 1 = (2^{m})^{k} - 1 = (2^{m}-1)(( 2^{m} )^{n-1} + ( 2^{m} )^{n-2} + \cdots + 1)$

  It is not a prime

  Contradiction

 

  $\therefore$  $n$ is a prime.