Prime constant

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The prime constant is the number ρ whose nth 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 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: ρ = 0.414682509851111660248109622...

[edit] Irrationality

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

Denote the kth digit of the binary expansion of ρ by rk. Then, since ρ is assumed rational, there must exist N, k positive integers such that rn = rn + 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 rp = 1. As noted, we have rp = rp + 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 ρ is irrational.