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{{Short description|Decimal number, 0.235711...}}
The '''Copeland-Erdős constant''' is the concatenation of "0." with the base 10 representations of the ]s in order. Its value is approximately The '''Copeland–Erdős constant''' is the concatenation of "0." with the ] representations of the ]s in order. Its value, using the modern definition of prime,<ref>Copeland and Erdős considered 1 a prime, and they defined the constant as 0.12357111317...</ref> is approximately


:0.235711131719232931374143... {{OEIS|id=A33308}} :0.235711131719232931374143... {{OEIS|id=A033308}}.


The constant is irrational. By ], there exist primes of the form The constant is ]; this can be proven with ] or ] (Hardy and Wright, p. 113) or ] that every ] ] is a sum of at most six primes. It also follows directly from its normality (see below).
:<math>k 10^m + 1</math>
for all positive integers <math>m</math>. Hence, there exist primes with digit strings containing arbitrarily long sequences of zeros followed by the digit 1. Thus, the digit string of the Copeland-Erdős constant contains arbitrarily long sequences of zeros
followed by the digit 1, and hence the digit string of the constant cannot terminate or recur. So, the constant is irrational (Hardy and Wright, p. 113).


By a similar argument, any constant created by concatenating "0." with all primes in an ] <math>d \cdot n + a</math>, where ''a'' is ] to ''d'' and to 10, will be irrational. E.g. primes of the form <math>4n+1</math> or <math>8n-1</math>. By Dirichlet's theorem, the arithmetic progression <math>d \cdot n \cdot 10^m + a</math> contains primes for all ''m'', and those primes are also in <math>d \cdot n + a</math>, so the concatenated primes contain arbitrarily long sequences of the digit zero. By a similar argument, any constant created by concatenating "0." with all primes in an ] ''dn''&nbsp;+&nbsp;''a'', where ''a'' is ] to ''d'' and to 10, will be irrational; for example, primes of the form 4''n''&nbsp;+&nbsp;1 or 8''n''&nbsp;+&nbsp;1. By Dirichlet's theorem, the arithmetic progression ''dn''&nbsp;· 10<sup>''m''</sup>&nbsp;+&nbsp;''a'' contains primes for all ''m'', and those primes are also in ''cd''&nbsp;+&nbsp;''a'', so the concatenated primes contain arbitrarily long sequences of the digit zero.


In base 10, the constant is a ], a fact proven by ] and ] in 1946 (hence the name of the constant). In base 10, the constant is a ], a fact proven by ] and ] in 1946 (hence the name of the constant).<ref>{{harvnb|Copeland|Erdős|1946}}</ref>


The constant is given by The constant is given by
:<math>\displaystyle \sum_{n=1}^\infty p(n) 10^{-\left(n + \sum_{k=1}^n \lfloor \log_{10}{p(n)} \rfloor \right)}</math> :<math>\displaystyle \sum_{n=1}^\infty p_n 10^{-\left(n + \sum_{k=1}^n \lfloor \log_{10}{p_k} \rfloor \right)}</math>
⚫ where p(n) gives the n-th ].


⚫ where ''p<sub>n</sub>'' is the ''n''th ].
⚫ Its ] is ({{OEIS2C|id=A30168}})


⚫ Its ] is ({{OEIS2C|id=A030168}}).
⚫ The larger ]s approximate the value of this constant multiplied by the appropriate power of 10.

==Related constants==

Copeland and Erdős's proof that their constant is normal relies only on the fact that <math>p_n</math> is ] and <math>p_n = n^{1+o(1)}</math>, where <math>p_n</math> is the ''n''<sup>th</sup> prime number. More generally, if <math>s_n</math> is any strictly increasing sequence of ]s such that <math>s_n = n^{1+o(1)}</math> and <math>b</math> is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the ]-<math>b</math> representations of the <math>s_n</math>'s is normal in base <math>b</math>. For example, the sequence <math>\lfloor n (\log n)^2\rfloor</math> satisfies these conditions, so the constant 0.003712192634435363748597110122136... is normal in base 10, and 0.003101525354661104...<sub>7</sub> is normal in base 7.

In any given base ''b'' the number

: <math>\displaystyle \sum_{n=1}^\infty b^{-p_n}, \, </math>

which can be written in base ''b'' as 0.0110101000101000101...<sub>''b''</sub>
where the ''n''th digit is 1 if and only if ''n'' is prime, is irrational.<ref>{{harvnb|Hardy| Wright|1979| p=112}}</ref>

==See also==
⚫ *]s: the truncated value of this constant multiplied by the appropriate power of 10.
*]: concatenating all natural numbers, not just primes.


==References== ==References==
{{reflist}}
===Sources===
*{{citation|last1=Copeland|first1= A. H.|last2= Erdős|author-link1=Arthur Herbert Copeland|first2= P. |author-link2=Paul Erdős|title=Note on Normal Numbers|journal= ]|volume= 52|pages= 857–860|date= 1946|issue= 10|doi=10.1090/S0002-9904-1946-08657-7|doi-access= free}}.
*{{citation|author1-link=G. H. Hardy|last1=Hardy|first1=G. H.|author2-link=E. M. Wright|first2=E. M.|last2=Wright|date=1979|orig-year=1938|title=An Introduction to the Theory of Numbers|publisher=Oxford University Press|edition=5th|isbn=0-19-853171-0|url-access=registration|url=https://archive.org/details/introductiontoth00hard}}.


==External links==
* ] and ] (]) ''An Introduction to the Theory of Numbers'', Oxford University Press, USA; 5th edition (April 17, 1980). ISBN 0-19-853171-0.
*{{MathWorld|title=Copeland-Erdos Constant|urlname=Copeland-ErdosConstant}} *{{MathWorld|title=Copeland-Erdos Constant|urlname=Copeland-ErdosConstant}}


{{DEFAULTSORT:Copeland-Erdos constant}}
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Latest revision as of 15:53, 11 November 2024

Decimal number, 0.235711...

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, 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 · 10 + 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

∑ 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 simple continued fraction is (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 n 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.

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

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