# 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 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).

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, …] ( A30168).

## Related constants

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 *n* is prime, is irrational. (Hardy and Wright, p. 112).

## See also

- Smarandache–Wellin numbers: the truncated value of this constant multiplied by the appropriate power of 10.

## References

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