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,
where p indicates a prime and 
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 
.
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 . Now consider the case i = p. We have 
, since p(k + 1) is composite because 
. Since 
we see that ρ is irrational.
