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Prime constant

The prime constant is the number $ρ$ whose $n$ th binary digit is 1 if $n$ is prime and 0 if it is composite.

In other words, $ρ$ is simply the number whose binary expansion corresponds to the characteristic function of the set of prime numbers. That is,

$\rho = \sum_{p} \frac{1}{2^p} = \sum_{n=1}^\infty \frac{\chi_{\mathbb{P}}(n)}{2^n}$

where $p$ indicates a prime and $\chi_{\mathbb{P}}$ is the characteristic function of the primes.

The beginning of the decimal expansion of ρ is: $ρ = 0.414682509851111660248109622...$

Irrationality

The number $ρ$ is easily shown to be irrational. To see why, suppose it were rational.

Denote the $k$ th digit of the binary expansion of $ρ$ by $r k$ . Then, since $ρ$ is assumed rational, there must exist $N$ , $k$ positive integers such that $r n = r n + i k$ 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 $r p = r p + i k$ 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 $ρ$ is irrational.

Categories: Irrational numbers

Last updated: 05-27-2005 00:09:46

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10-26-2009 08:16:03
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