Just curious about the following problem:
Show that $\sum\limits_{p}{\frac{1}{2^p}}$ where $p$ runs over prime numbers is an irrational number.
It goes under 'Pigeonhole Principle' topic in one of problem set I found on the Web. My idea was to assume $S=\frac{p}{q}$ and show that $q$ can be higher than any $N$. But I can't see any way to implement it.
Here is an approach suggested by achille hui.
Suppose $s = \sum\limits_{p}{\frac{1}{2^p}}$ is rational, i.e a periodic binary fraction.
Since each $\frac{1}{2^p}$ as a binary fraction has 1 only in $p-th$ position, the sequence of prime numbers must be periodic. In other words, if $n$ is the length of period for $s$ then it should be a prime number $p$ so that $p+nk$ is a prime number for any $k$. But $p+np$ is a composite number.