The set $S$ of even perfect numbers $n$ such that $n+1$ is a prime number contains
$$
6,28,33550336,137438691328
$$


Latter number found by Joerg Arndt, corresponds to $M\_{19}$ (mersenne)


Question: Is $S$ reduced to these $4$ numbers.


New: Joerg Arndt checked up to exponent $110503$ that the corresponding number $n+1$
is composite. (Improved $19$ to $110503$).


Which function of $x$ migh describe `well` the size of the set of elements in $S$ less than $x$


divided


by the size of the set of all even perfect numbers less than $x$; mainly with `big` $x.$


So, I am asking for `relative` size not `absolute` size. E.g., if I were asking
for relative density of the prime numbers congruent to $3$ modulo $4$: I do not want to use
the big machinery of the prime number theorem, or Dirichlet's Theorem to deduce how many should be there. I just want (in these case) to know how to describe in terms of $x$


number of primes congruent to $3$ modulo $4$ and less than $x$


divided by


number of primes less than $x$


How many such numbers $n$ we may
expect inside the known 47 perfect numbers ?



Even Perfect numbers $n$ with $n+1$ prime
There's a conjecture (for which I can't find a source now) that the number of Mersenne primes $2^n-1$ with $n < x$ is $c \log x$ for some constant $c$. Differentiating this, the "probability" that $2^n-1$ is prime is about $c/n$. (This is unconditional; that is, I'm not assuming $n$ is prime.) 


The even perfect numbers are exactly of the form $2^{n-1}(2^n-1)$ with $2^n-1$ prime.


So the "probability" that $2^n-1$ and $2^{n-1} (2^n-1) + 1$ is prime, assuming independence, is $c/n$ times the probability that $2^{n-1} (2^n-1) + 1$ is prime. $2^{n-1} (2^n-1) + 1$ is roughly $2^{2n}$, so by the prime number theorem its "probability" of being prime is about $1/log(2^{2n})$, or again a constant divided by $n$. That is, the "probability" that $2^n-1$ is prime and the corresponding number is one less than a prime is $c/n^2$; since $\sum\_{n \ge 1} cn^{-2}$ is finite this leads us to suspect that there are finitely many solutions.


Of course none of this is anywhere near being a proof...


