The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value, using the modern definition of prime,[1] is approximately
0.235711131719232931374143... (sequence A033308 in the OEIS).
By a similar argument, any constant created by concatenating "0." with all primes in an arithmetic progressiondn + a, where a is coprime to d and to 10, will be irrational; for example, 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.
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.[3]
See also
Smarandache–Wellin numbers: the truncated value of this constant multiplied by the appropriate power of 10.