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. E.g. 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

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

where p_n is the nth prime number.

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

In any given base b the number

${\displaystyle \displaystyle \sum _{n=1}^{\infty }b^{-p_{n}},\,}$

which can be written in base b as 0.0110101000101000101…_b where the nth 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

1. Copeland and Erdős considered 1 a prime, and they defined the constant as 0.12357111317…
2. Copeland & Erdős 1946
3. 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.

External links

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