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 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.
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).[1]
The constant is given by
where pn is the nth prime number.
Its continued fraction is [0; 4, 4, 8, 16, 18, 5, 1, …] (
Related constants
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.[2]
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.
References
Sources
- 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) [1938], An Introduction to the Theory of Numbers (5th ed.), Oxford University Press, ISBN 0-19-853171-0 .