The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value is approximately
The constant is irrational; this can be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate (Hardy and Wright, p. 113) or Ramare's theorem that every even integer is a sum of at most six primes. It also follows directly from its normality (see below).
By a similar argument, any constant created by concatenating "0." with all primes in an arithmetic progression dn + a, where a is coprime to d and to 10, will be irrational. E.g. primes of the form 4n + 1 or 8n + 1. By Dirichlet's theorem, the arithmetic progression dn·10m + a contains primes for all m, and those primes are also in cd + a, so the concatenated primes contain arbitrarily long sequences of the digit zero.
The constant is given by
where pn is the nth prime number.
In any given base b the number
which can be written in base b as 0.0110101000101000101…b where the nth digit is 1 if n is prime, is irrational. (Hardy and Wright, p. 112).