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
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
- [math]\displaystyle{ \displaystyle \sum_{n=1}^\infty p_n 10^{-\left(n + \sum_{k=1}^n \lfloor \log_{10}{p_k} \rfloor \right)} }[/math]
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 [math]\displaystyle{ p_n }[/math] is strictly increasing and [math]\displaystyle{ p_n = n^{1+o(1)} }[/math], where [math]\displaystyle{ p_n }[/math] is the nth prime number. More generally, if [math]\displaystyle{ s_n }[/math] is any strictly increasing sequence of natural numbers such that [math]\displaystyle{ s_n = n^{1+o(1)} }[/math] and [math]\displaystyle{ b }[/math] is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the base-[math]\displaystyle{ b }[/math] representations of the [math]\displaystyle{ s_n }[/math]'s is normal in base [math]\displaystyle{ b }[/math]. For example, the sequence [math]\displaystyle{ \lfloor n (\log n)^2\rfloor }[/math] 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
- [math]\displaystyle{ \displaystyle \sum_{n=1}^\infty b^{-p_n}, \, }[/math]
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
- ↑ Copeland and Erdős considered 1 a prime, and they defined the constant as 0.12357111317…
- ↑ Copeland & Erdős 1946
- ↑ Hardy & Wright 1979, p. 112
Sources
- Copeland, A. H.; Erdős, P. (1946), "Note on Normal Numbers", Bulletin of the American Mathematical Society 52 (10): 857–860, doi:10.1090/S0002-9904-1946-08657-7.
- Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (5th ed.), Oxford University Press, ISBN 0-19-853171-0, https://archive.org/details/introductiontoth00hard.
External links
- Weisstein, Eric W.. "Copeland-Erdos Constant". http://mathworld.wolfram.com/Copeland-ErdosConstant.html.
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