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| The '''Copeland–Erdős constant''' is the concatenation of "0." with the base 10 representations of the [[prime number]]s in order. Its value is approximately
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| :0.235711131719232931374143… {{OEIS|id=A33308}}. | |
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| 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 [[Olivier Ramaré|Ramare's theorem]] that every even integer is a sum of at most six primes. It also follows directly from its normality (see below).
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| 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 4''n'' + 1 or 8''n'' + 1. By Dirichlet's theorem, the arithmetic progression ''dn''·10<sup>''m''</sup> + ''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.
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| 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).
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| The constant is given by
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| :<math>\displaystyle \sum_{n=1}^\infty p_n 10^{-\left(n + \sum_{k=1}^n \lfloor \log_{10}{p_k} \rfloor \right)}</math>
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| where ''p<sub>n</sub>'' is the ''n''th [[prime number]].
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| Its [[continued fraction]] is [0; 4, 4, 8, 16, 18, 5, 1, …] ({{OEIS2C|id=A30168}}).
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| ==Related constants==
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| In any given base ''b'' the number
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| : <math>\displaystyle \sum_{n=1}^\infty b^{-p_n}, \, </math> | |
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| which can be written in base ''b'' as 0.0110101000101000101…<sub>''b''</sub>
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| where the ''n''th digit is 1 if ''n'' is prime, is irrational. (Hardy and Wright, p. 112).
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| ==See also==
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| *[[Smarandache–Wellin number]]s: the truncated value of this constant multiplied by the appropriate power of 10.
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| ==References==
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| *{{citation|author1-link=G. H. Hardy|last1=Hardy|first1=G. H.|author2-link=E. M. Wright|first2=E. M.|last2=Wright|year=1938|title=An Introduction to the Theory of Numbers|publisher=Oxford University Press|edition=5th|isbn=0-19-853171-0}}.
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| ==External links==
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| *{{MathWorld|title=Copeland-Erdos Constant|urlname=Copeland-ErdosConstant}}
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| {{DEFAULTSORT:Copeland-Erdos constant}}
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| [[Category:Irrational numbers]]
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| [[Category:Prime numbers]]
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