Prime constant

The prime constant is the real number

th binary digit is 1 if

is composite or 1.

[1] In other words,

is the number whose binary expansion corresponds to the indicator function of the set of prime numbers.

indicates a prime and

χ

is the characteristic function of the set

of prime numbers.

The beginning of the decimal expansion of ρ is:

ρ = 0.414682509851111660248109622 …

(sequence A051006 in the OEIS)[1] The beginning of the binary expansion is:

ρ = 0.011010100010100010100010000

(sequence A010051 in the OEIS) The number

is irrational.

were rational.

Denote the

th digit of the binary expansion of

is assumed rational, its binary expansion is eventually periodic, and so there exist positive integers

Since there are an infinite number of primes, we may choose a prime

By definition we see that

As noted, we have

Now consider the case

is composite because

is irrational.