Prime constant

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The prime constant is the real number \rho whose nth binary digit is 1 if n is prime and 0 if n is composite or 1.

In other words, \rho is simply the number whose binary expansion corresponds to the indicator 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:  \rho = 0.414682509851111660248109622\ldots (sequence A051006 in OEIS)

The beginning of the binary expansion is:  \rho = 0.011010100010100010100010000\ldots_2 (sequence A010051 in OEIS)


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

Denote the kth 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.