# prime constant

The number $\rho$ defined by

 $\rho=\sum_{p}\frac{1}{2^{p}}$

is known as the prime constant. It is simply the number whose binary expansion corresponds to the characteristic function of the set of prime numbers. That is, its $n$th binary digit is $1$ if $n$ is prime and $0$ if $n$ is composite.

The beginning of the decimal expansion of $\rho$ is:

 $\rho=0.414682509851111660248109622...$

The number $\rho$ is easily shown to be irrational. To see why, suppose it were rational. Denote the $k$th digit of the binary expansion of $\rho$ by $r_{k}$. Then, since $\rho$ is assumed rational, there must exist $N$, $k$ positive integers such that $r_{n}=r_{n+ik}$ for all $n>N$ and all $i\in\mathbb{N}$.

Since there are an infinite number of primes, we may choose a prime $p>N$. By definition we see that $r_{p}=1$. As noted, we have $r_{p}=r_{p+ik}$ for all $i\in\mathbb{N}$. Now consider the case $i=p$. We have $r_{p+i\cdot k}=r_{p+p\cdot k}=r_{p(k+1)}=0$, since $p(k+1)$ is composite because $k+1\geq 2$. Since $r_{p}\neq r_{p(k+1)}$ we see that $\rho$ is irrational.

The partial continued fractions of the prime constant can be found http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A051007here.

Title prime constant PrimeConstant 2013-03-22 15:02:17 2013-03-22 15:02:17 mathcam (2727) mathcam (2727) 12 mathcam (2727) Definition msc 11A41