The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value is approximately

0.235711131719232931374143… (sequence A33308 in the OEIS).

The constant can be proved to be irrational by the use of either Dirichlet's theorem on arithmetic progressions or Chebyshev's theorem. (Hardy and Wright, p. 113).

By a similar argument, any constant created by concatenating "0." with all primes in an arithmetic progression , where a is coprime to d and to 10, will be irrational. E.g. primes of the form or . By Dirichlet's theorem, the arithmetic progression contains primes for all m, and those primes are also in , 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) gives the n-th prime number.

Its continued fraction is [0; 4, 4, 8, 16, 18, 5, 1, …] (OEIS: A30168).

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

• Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (5th ed.), Oxford University Press, ISBN 0-19-853171-0.

• Weisstein, Eric W. "Copeland-Erdos Constant". MathWorld.