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The prime constant is the real number ρ {\displaystyle \rho } whose n {\displaystyle n} th binary digit is 1 if n {\displaystyle n} is prime and 0 if n {\displaystyle n} is composite or 1.
In other words, ρ {\displaystyle \rho } is the number whose binary expansion corresponds to the indicator function of the set of prime numbers. That is,
where p {\displaystyle p} indicates a prime and χ P {\displaystyle \chi _{\mathbb {P} }} is the characteristic function of the set P {\displaystyle \mathbb {P} } of prime numbers.
The beginning of the decimal expansion of ρ is: ρ = 0.414682509851111660248109622 … {\displaystyle \rho =0.414682509851111660248109622\ldots }