# 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:

$$\rho =0.414682509851111660248109622\mathrm{\dots}$$ |

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 $k+1\ge 2$. Since ${r}_{p}\ne {r}_{p(k+1)}$ 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 |
---|---|

Canonical name | PrimeConstant |

Date of creation | 2013-03-22 15:02:17 |

Last modified on | 2013-03-22 15:02:17 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 12 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11A41 |