# Mersenne primes

A Mersenne prime is a prime number of the form $2^p-1$ where $p$ has to be a prime number. Now, let $p_0$ be a prime number, and let us define the sequence $p_n = 2^{p_{n-1}}-1$. Is there a $p_0$ such that the sequence $p_n$ is a sequence of primes ? I checked it for the first 4 terms (starting with $p_0 = 2$) and it looks like it works. Obviously I'm sure it fails at some point for such simple case, but is there a $p_0$ such that this sequence is a sequence of prime numbers ?

Edit: Funny, it works for the 5 first terms starting at $p_0 = 2$

In particular, $p_{n+1} = p_n + \sum_{k = p_{n-1}}^{p_{n} - 1} 2^k$

Another fun formula is $p_{n+1} +1 = (p_n +1) 2^{p_n - p_{n-1}}$, or simply $\prod_{k=1}^n (1+p_k) = 2^{\sum_{k=0}^{n-1} p_k}$, that is, $\sum_{k=0}^{n-1} p_k = \sum_{k=1}^n ln(1+ p_k)/ln(2)$. You have so many of them because $1+p_{n+1} = 2^{p_n}$ and $p_{n+1} = \sum_{k=0}^{p_n-1} 2^{k}$ :<

• It is not known if/that there are infinitely many Mersenne primes so such a sequecne is definitely not known. It might be possible to give an argument why it cannot exist. But I do not know one.
– quid
Nov 11 '14 at 23:43