Copeland–Erdős constant explained

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... .

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).

The constant is given by

\displaystyle \sum ^{inf ty} _{n =1} p _{n} 10 ^{- \left(n + \sum ^{n} _{k =1} \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, ...] .

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.

In any given base b the number

\displaystyle \sum ^{inf ty} _{n =1} b ^{- p _{n}} ,

which can be written in base b as 0.0110101000101000101..._bwhere the nth digit is 1 if and only if n is prime, is irrational.

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

Sources

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Notes and References

1. Copeland and Erdős considered 1 a prime, and they defined the constant as 0.12357111317...