The prime constant is the real number ρ whose n th 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,
ρ = ∑ p 1 2 p = ∑ n = 1 ∞ χ P ( n ) 2 n where p indicates a prime and χ P is the characteristic function of the primes.
The beginning of the decimal expansion of ρ is: ρ = 0.414682509851111660248109622 … (sequence A051006 in the OEIS)
The beginning of the binary expansion is: ρ = 0.011010100010100010100010000 … 2 (sequence A010051 in the OEIS)
A close approximation can be obtained using the golden ratio and the polygon circumscribing constant:
φ 5 / κ The prime constant has been used in the calculation of value of the inverse fine-structure constant, together with traditional Pythagorean triangles:
α − 1 ≈ 157 − 337 ρ / 7 The number ρ is easily shown to be irrational. To see why, suppose it were rational.
Denote the k th 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 + i k for all n > N and all i ∈ 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 + i k for all i ∈ N . Now consider the case i = p . We have r p + i ⋅ k = r p + p ⋅ k = r p ( k + 1 ) = 0 , since p ( k + 1 ) is composite because k + 1 ≥ 2 . Since r p ≠ r p ( k + 1 ) we see that ρ is irrational.
