# 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,[1] 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 4*n* + 1 or 8*n* + 1. By Dirichlet's theorem, the arithmetic progression *dn*·10^{m} + *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).[2]

The constant is given by

where *p _{n}* is the

*n*th prime number.

Its continued fraction is [0; 4, 4, 8, 16, 18, 5, 1, …] (OEIS: A030168).

## Related constants

Copeland and Erdős's proof that their constant is normal relies only on the fact that is strictly increasing and , where is the n^{th} 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 *n*th 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.
- Champernowne constant: concatenating all natural numbers, not just primes.

## References

- Copeland and Erdős considered 1 a prime, and they defined the constant as 0.12357111317…
- Copeland & Erdős 1946
- Hardy & Wright 1979, p. 112

### 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.