Copeland–Erdős constant
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, 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; for example, primes of the form 4"n" + 1 or 8"n" + 1. By Dirichlet's theorem, the arithmetic progression "dn" · 10 + "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. In base 10, the constant is a normal number, a fact proven by Arthur Herbert Copeland and Paul Erdős in 1946 (hence the name of the constant). The constant is given by where "p" is the  nth prime number. Its continued fraction is [0; 4, 4, 8, 16, 18, 5, 1, …] (). Related constants. Copeland and Erdős's proof that their constant is normal relies only on the fact that formula_2 is strictly increasing and formula_3, where formula_2 is the "n" prime number. More generally, if formula_5 is any strictly increasing sequence of natural numbers such that formula_6 and formula_7 is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the base-formula_7 representations of the formula_5's is normal in base formula_7. For example, the sequence formula_11 satisfies  these conditions, so the constant 0.003712192634435363748597110122136… is normal in base 10, and 0.003101525354661104… is normal in base 7. In any given base "b" the number which can be written in base "b" as 0.0110101000101000101… where the "n"th digit is 1 if and only if "n" is prime, is irrational.