Mersenne Prime - why are these two definitions equivalent? According to Wikipedia:

If $n$ is a composite number then so is $2^n − 1$. ($2^{ab} − 1$ is divisible
  by both $2^a − 1$ and $2^b − 1$.) This definition is therefore equivalent to
  a definition as a prime number of the form $M_p = 2^p − 1$ for some prime
  $p$.

I'm wondering why the definition is equivalent. If $p$ is prime, it doesn't necessarily mean that $2^p-1$ is prime, of course. It does, however, mean from my understanding that by the contrapositive, if $2^n-1$ is prime, then $n$ is prime, but that doesn't help much. So what precisely does the second sentence mean?
 A: It is not clear from your extract which definitions are being counted as equivalent. It looks like:
A Mersenne Prime is a prime of the form $2^n-1$
and
A Mersenne Prime is a prime of the form $2^p-1$ with $p$ prime
And the fact that if $n$ is composite, so is $2^n-1$ means that these define the same set of primes.
A: The claim is that the following two are equivalent for each $N$:


*

*$N$ is a prime and $N$ can be written as $N=2^n-1$ where $n$ is a natural number.

*$N$ is a prime and $N$ can be written as $N=2^p-1$ where $p$ is a prime.


Further, they say that because they are equivalent, both can be used as the definition of a Mersenne prime.
Proof that they are equivalent:


*

*Proof that 2 implies 1: Assume 2 is true. Just set $n=p$ in 1. Done.

*Proof that 1 implies 2: Assume 1 is true. $N=2^n-1$ is a prime, so $n$ can't be composite (as shown in your quote). So $n$ is a prime. So $N$ can be written as $2^p-1$ where $p$ is a prime. So 2 is true.

A: As stated, a Mersenne prime is a Mersenne number, i.e., a number of the form
$$M_n=2^n-1$$
which is itself prime. Thus, in order for $M_n$ to be prime,  $n$ must also be prime. This follows, as for any composite $n$ with factors $a$ and $b$, $n=ab$. 
Thus $2^n-1$ can be written as $2^{ab}-1$, which is a binomial number that always has a factor  $(2^a-1)$.
A: I think the paragraph is trying to say the following for $M_n$
$$
n \textrm{ composite} \Rightarrow M_n  \textrm{ composite}
$$
is equivalent to
$$
M_n \neg  \textrm{ composite} 
\Rightarrow n \neg
 \textrm{ composite}
$$
I.e. is equivalent to
$$
M_n \textrm{ prime} 
\Rightarrow n 
 \textrm{ prime}
$$
