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.
Copeland and Erdős's proof that their constant is normal relies only on the fact that is strictly increasing and , where is the nth prime number. More generally, if is any strictly increasing sequence of natural numbers such that and is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the base- representations of the 's is normal in base . For example, the sequence satisfies these conditions, so the constant 0.003712192634435363748597110122136… is normal in base 10, and 0.003101525354661104…7 is normal in base 7.
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 and only if n is prime, is irrational.
- Smarandache–Wellin numbers: the truncated value of this constant multiplied by the appropriate power of 10.
- Champernowne constant: concatenating all natural numbers, not just primes.
- Copeland, A. H.; Erdős, P. (1946), "Note on Normal Numbers", Bulletin of the American Mathematical Society, 52: 857–860, doi:10.1090/S0002-9904-1946-08657-7.
- Hardy, G. H.; Wright, E. M. (1979) , An Introduction to the Theory of Numbers (5th ed.), Oxford University Press, ISBN 0-19-853171-0.