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 · 10m + 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

∑ n = 1 ∞ p n 10 − ( n + ∑ k = 1 n ⌊ log 10 ⁡ p k ⌋ ) {\displaystyle \displaystyle \sum _{n=1}^{\infty }p_{n}10^{-\left(n+\sum _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor \right)}}

where pn 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 p n {\displaystyle p_{n}} is strictly increasing and p n = n 1 + o ( 1 ) {\displaystyle p_{n}=n^{1+o(1)}} , where p n {\displaystyle p_{n}} is the nth prime number. More generally, if s n {\displaystyle s_{n}} is any strictly increasing sequence of natural numbers such that s n = n 1 + o ( 1 ) {\displaystyle s_{n}=n^{1+o(1)}} and b {\displaystyle b} is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the base- b {\displaystyle b} representations of the s n {\displaystyle s_{n}} 's is normal in base b {\displaystyle b} . For example, the sequence ⌊ n ( log ⁡ n ) 2 ⌋ {\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

∑ n = 1 ∞ b − p n , {\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

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

External links