Prime constant
The prime constant is the real number formula_1 whose formula_2th binary digit is 1 if formula_2 is prime and 0 if formula_2 is composite or 1. In other words, formula_1 is the number whose binary expansion corresponds to the indicator function of the set of prime numbers. That is, where formula_7 indicates a prime and formula_8 is the characteristic function of the set formula_9 of prime numbers. The beginning of the decimal expansion of "ρ" is: formula_10 The beginning of the binary expansion is: formula_11 Irrationality. The number formula_1 can be shown to be irrational. To see why , suppose it were rational. Denote the formula_13th digit of the binary expansion of formula_1 by formula_15. Then since formula_1 is assumed rational, its binary expansion is periodic, and so there exist positive integers formula_17 and formula_13 such that formula_19 for all formula_20 and all formula_21. Since there are an infinite number of primes, we may choose a prime formula_22. By definition we see that formula_23. As noted, we have formula_24 for all formula_21. Now consider the case formula_26. We have formula_27, since formula_28 is composite because formula_29. Since formula_30 we see that formula_1 is irrational.