en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T09:53:53Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 11:53, 5 May 2014
Line 1:		Line 1:
	~~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~~		Hi there. My title [http://www.Familysurvivalgroup.com/easy-methods-planting-looking-backyard/ psychic readings] is Sophia Meagher even though it is not the name on my beginning certification. To play lacross is [http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies psychic love readings] the thing I adore most of all. Since I was 18 I've been operating as a bookkeeper but quickly my spouse and I will begin our own business. Her family life in Ohio.<br><br>Here is my website clairvoyant psychic ([http://www.weddingwall.com.au/groups/easy-advice-for-successful-personal-development-today/ please click the following webpage])

	:0.~~235711131719232931374143… {{OEIS\|id=A33308}}~~.

	~~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).~~

	~~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~~.

	~~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~~
	~~:<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~~<~~sub~~>n<~~/sub~~>'' is ~~the ''n''th~~ [~~[prime number]].~~

	~~Its [[continued fraction]] is [0; 4, 4, 8, 16, 18, 5, 1, …] ({{OEIS2C\|id=A30168}}).~~

	~~==Related constants==~~

	~~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 ''n'' is prime, is irrational~~. ~~(Hardy and Wright, p~~.~~ 112)~~.

	~~==See also==~~
	*[[Smarandache–Wellin number]]s: the truncated value of this constant multiplied by the appropriate power of 10.

	~~==References==~~
	*{{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}}.~~

	~~==External links==~~
	*{{MathWorld\|title=Copeland-~~Erdos Constant\|urlname=Copeland-ErdosConstant}}~~

	~~{{DEFAULTSORT:Copeland-Erdos constant}}~~
	~~[[Category:Irrational numbers]]~~
	~~[[Category:Prime numbers]~~]

en>Omnipaedista: noref-tag

2011-05-21T04:47:56Z

noref-tag

New page

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

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

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).

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''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
:<math>\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 ''n''th [[prime number]].

Its [[continued fraction]] is [0; 4, 4, 8, 16, 18, 5, 1, …] ({{OEIS2C|id=A30168}}).

==Related constants==

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…''b''
where the ''n''th digit is 1 if ''n'' is prime, is irrational. (Hardy and Wright, p. 112).

==See also==
*[[Smarandache–Wellin number]]s: the truncated value of this constant multiplied by the appropriate power of 10.

==References==
*{{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}}.

==External links==
*{{MathWorld|title=Copeland-Erdos Constant|urlname=Copeland-ErdosConstant}}

{{DEFAULTSORT:Copeland-Erdos constant}}
[[Category:Irrational numbers]]
[[Category:Prime numbers]]

Difference hierarchy - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

en>Omnipaedista: noref-tag