The number defined by
is known as the prime constant. It is simply the number whose binary expansion corresponds to the characteristic function of the set of prime numbers. That is, its th binary digit is if is prime and if is composite.
The beginning of the decimal expansion of is:
The number is easily 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, there must exist , positive integers 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.
|Date of creation||2013-03-22 15:02:17|
|Last modified on||2013-03-22 15:02:17|
|Last modified by||mathcam (2727)|