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 is irrational. By Dirichlet's theorem on arithmetic progressions, there exist primes of the form

for all positive integers and . Hence, there exist primes with digit strings containing arbitrarily long sequences of zeros followed by the digit 1. Thus, the digit string of the Copeland-Erdős constant contains arbitrarily long sequences of zeros followed by the digit 1, and hence the digit string of the constant cannot terminate or recur. So, the constant is irrational (Hardy and Wright, p. 113).

By a simlar argument, any constant created by concatenating all primes in a given arithmetic progression (e.g., primes of the form or ) will be irrational.

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)

The larger Smarandache-Wellin numbers approximate the value of this constant multiplied by the appropriate power of 10.

• Hardy G. H. and E. M. Wright (1938) An Introduction to the Theory of Numbers, Oxford University Press, USA; 5th edition (April 17, 1980). ISBN 0-19-853171-0.