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 can be shown to be irrational. To see why, suppose it 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. References External links Irrational numbers Prime numbers Articles containing proofs Mathematical constants