Prime constant

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

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: $\rho = 0.414682509851111660248109622\ldots$ (sequence A051006 in OEIS)

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

Irrationality

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 r_k. Then, since ρ 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 ρ is irrational.

References

• Weisstein, Eric W., "Prime Constant" from MathWorld.