In mathematics, 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
. In base 10, the constant is a normal number, a fact proven by Arthur Herbert Copeland and Paul Erdős. It follows that the constant is irrational; this can also be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate or Ramaré's theorem that every even integer is a sum of at most six primes.
By a similar argument, any constant created by concatenating "0." with all primes in an arithmetic progression , where is coprime to and to 10, will be irrational. By Dirichlet's theorem, the arithmetic progression contains primes for all , and those primes are also in , so the concatenated primes contain arbitrarily long sequences of the digit zero.
Its simple continued fraction is .
Related constants
Copeland and Erdős's proof that their constant is normal relies only on the fact that the sequence of primes is strictly increasing and . 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 terms is normal in base . For example, the sequence satisfies these conditions, so the constant
is normal in base 10, and is normal in base 7.
Moreover, in any given base the number , which can be written in base as
where the th digit is 1 if and only if is prime, is irrational.
See also 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
Notes
References
Sources
External links