Mersenne Prime - why are these two definitions equivalent?
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.
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.