TITLE: $\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number
QUESTION [16 upvotes]: I need help to prove the following result.

$\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.

REPLY [6 votes]: One way to show that the sum is irrational
is the well-known fact that there
are arbitrarily long intervals
where there are no primes.
If the sum were rational,
chose the interval longer that the period,
and hello contradiction!
In particular,
$n!+k$ for $k = 2$ to $n$
has no primes, since each element is divisible
by a prime from $2$ to $n$.
This also shows that
$\sum_{\text{prime }p} m^{-p}$
is irrational for any integer $m \ge 2$.