Prime constant

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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)

Irrationality[edit]

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.

Notes[edit]


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