|This article needs additional citations for verification. (May 2011)|
where indicates a prime and is the characteristic function of the primes.
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.