Section 4.6 Mersenne Primes
Theorem 4.6.1.
If \(m \gt 1\) and \(a^m- 1\) is prime, then \(a = 2\) and \(m\) is prime.
Proof.
If \(a \gt 2\) then \(a^m-1 = (a-1)(1 + a +... + a^{m-1})\) is composite. Hence \(a = 2.\) If \(m = rs\) with \(r, s \gt 1\) then
\begin{equation*}
a^m-1 = (a^r)^s-1=(a^r-1)(1 + (a^r)+(a^r)^2+...+(a^r)^{s-1})
\end{equation*}
is composite. Hence \(m\) is prime.
Definition 4.6.2.
For \(n\in\mathbb{N}_0\) we call the number \(M_n = 2^n-1\) the \(n\)th Mersenne numbers. Those which are prime are called Mersenne prime.
Conjecture 4.6.3.
There are infinitely many Mersenne primes.
There is not yet a good approach to this conjecture. There are large computer searches going on and if you have a computer you can join in, see the site https://www.mersenne.org.
Remark 4.6.4.
The \(43\)rd Mersenne prime is \(9,152,052\) digits long.