Prime constant

The prime constant is the real number whose th binary digit is 1 if is prime and 0 if is composite or 1.
In other words, is the number whose binary expansion corresponds to the indicator function of the set of prime numbers. That is,
where indicates a prime and is the characteristic function of the set of prime numbers.
The beginning of the decimal expansion of ρ is:
The beginning of the binary expansion is:

Irrationality

The number is irrational.

Proof by contradiction

Suppose were rational.
Denote the th digit of the binary expansion of by. Then since is assumed rational, its binary expansion is eventually periodic, and so there exist positive integers and such that
for all and all.
Since there are an infinite number of primes, we may choose a prime. By definition we see that. As noted, we have for all. Now consider the case. We have, since is composite because. Since we see that is irrational.