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

0.235711131719232931374143... (sequence A033308 in the OEIS).

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; for example, primes of the form 4n + 1 or 8n + 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

\displaystyle \sum _{n=1}^{\infty }p_{n}10^{-\left(n+\sum _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor \right)}

where p_n is the nth prime number.

Its simple 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 $p_{n}$ is strictly increasing and $p_{n}=n^{1+o(1)}$ , where $p_{n}$ is the n^th prime number. More generally, if $s_{n}$ is any strictly increasing sequence of natural numbers such that $s_{n}=n^{1+o(1)}$ and $b$ is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the base- $b$ representations of the $s_{n}$ 's is normal in base $b$ . For example, the sequence $\lfloor n(\log n)^{2}\rfloor$ satisfies these conditions, so the constant 0.003712192634435363748597110122136... is normal in base 10, and 0.003101525354661104...₇ is normal in base 7.