Atmospheric refraction - Revision history

en>Tom.Reding: Gen fixes (page/s, endash, , et al., etc.) using AWB

2014-12-23T18:48:37Z

Gen fixes (page/s, endash, &nbsp, et al., etc.) using AWB

← Older revision		Revision as of 20:48, 23 December 2014
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	~~In [[number theory]], '''Mills' constant''' is defined as the smallest positive [[real number]] ''A'' such that the [[floor function\|floor]] of the [[double exponential function]]~~		Hello buddy. Allow me introduce myself. I am Luther Aubrey. Meter reading is exactly where my main earnings comes from but soon I'll be on my personal. Some time in the past he chose to reside in Kansas. The favorite pastime for my kids and me is dancing and now I'm trying to earn money with it.<br><br>Also visit my weblog - [http://togetherontime.com/UserProfile/tabid/61/userId/6564/Default.aspx extended auto warranty companies]

	~~:<math> \lfloor A^{3^{n}} \rfloor</math>~~

	~~is a [[prime number]], for all positive integers ''n''~~. ~~This constant is named after [[William H~~. ~~Mills]] who proved in 1947 the existence of ''A'' based on results of [[Guido Hoheisel]] and [[Albert Ingham]] on the [[prime gap]]s~~.
	~~Its value~~ is ~~unknown,~~ but ~~if the [[Riemann hypothesis]] is true it is approximately 1.3063778838630806904686144926... {{OEIS\|A051021}}.~~

	~~==Mills primes==~~
	~~The primes generated by Mills~~' ~~constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins~~
	~~:2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, .~~.~~. {{OEIS\|A051254}}.~~

	~~If ''a<sub>i</sub>'' denotes the ''i''<sup>th</sup> prime~~ in ~~this sequence, then ''a<sub>i</sub>'' can be calculated as~~ the ~~smallest prime number larger than <math>a_{i-1}^3</math>. In order~~ to ~~ensure that rounding <math>A^{3^n}</math>, for ''n'' = 1, 2, 3, …, produces this sequence of primes, it must be the case that <math>a_i < (a_{i-1}+1)^3</math>~~. The ~~Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large [[cubic number]]s, which~~ is sufficient to prove this inequality if we start from a sufficiently large first prime <math>a_1</math>. The Riemann hypothesis implies that there exists a prime between any two consecutive cubes, allowing the ''sufficiently large'' condition to be removed, and ~~allowing the sequence of Mills~~' ~~primes~~ to ~~begin at ''a''<sub>1</sub> = 2.~~

	~~Currently, the largest known Mills prime (under the Riemann hypothesis) is~~
	~~:<math>\displaystyle (((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220,</math>~~
	~~which is 20,562 digits long~~.

	~~==Numerical calculation==~~
	~~By calculating the sequence of Mills primes, one can approximate Mills' constant as~~
	:<~~math~~>~~A\approx a(n)^{1/3^n}.~~<~~/math~~>
	~~{{harvtxt\|Caldwell\|Cheng\|2005}} used this method to compute almost seven thousand base 10 digits of Mills' constant under the assumption that the [[Riemann hypothesis]] is true. There is no closed~~-~~form formula known for Mills' constant, and it is not even known whether this number is~~ [~~[rational number\|rational]] {{harv\|Finch\|2003}}.~~

	~~==See also==~~
	*[[Formula for primes]]

	~~==References==~~
	*{{citation\|last1=Caldwell\|first1=Chris K.\|last2=Cheng\|first2=Yuanyou\|title=Determining Mills' Constant and a Note on Honaker's Problem\|journal=Journal of Integer Sequences\|volume=8\|year=2005\|issue=5.4.1\|url=http://~~www~~.~~cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html}}.~~

	*{{citation\|first=Steven R.\|last=Finch\|title=Mathematical Constants\|year=2003\|publisher=Cambridge University Press\|isbn=0-521-81805-2\|contribution=Mills' Constant\|pages=130–133\|url=ftp://s208.math.msu.su/469000/dbcd69f8d83a96354dd49d21572c6432}}.

	*{{citation\|first=W. H.\|last=Mills\|title=A prime-representing function\|journal=[[Bulletin of the American Mathematical Society]]\|volume=53\|year=1947\|page=604\|doi=10.1090/S0002-9904-1947-08849-2\|issue=6\|url = http://~~www.ams.org~~/~~journals~~/~~bull~~/~~1947-53-06~~/~~S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf}}~~.

	~~==External links==~~
	* {{MathWorld\|urlname=MillsConstant\|title=Mills' Constant}}
	* [http://blogs.ethz.ch/kowalski/2009/04/02/who-remembers-the-mills-number/ Who remembers the Mills number?], E. Kowalski.
	* [http://www.youtube.com/watch?v=6ltrPVPEwfo Awesome Prime Number Constant], Numberphile.

	~~{{Prime number classes}}~~

	~~[[Category:Mathematical constants]]~~
	~~[[Category:Prime numbers]~~]

en>ClueBot NG: Reverting possible vandalism by 14.98.227.222 to version by 128.32.104.161. False positive? Report it. Thanks, ClueBot NG. (1632868) (Bot)

2013-12-27T09:04:43Z

Reverting possible vandalism by 14.98.227.222 to version by 128.32.104.161. False positive? Report it. Thanks, ClueBot NG. (1632868) (Bot)

← Older revision		Revision as of 11:04, 27 December 2013
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	~~Hi!~~ <br>~~My name~~ is ~~Alex~~ and I'm a ~~22 years old boy~~ from ~~Netherlands~~.<br><br>~~Feel free~~ to ~~visit my webpage~~ [http://~~Horochicago~~.com/~~WP/~~?~~attachment_id~~=~~62 Fifa coin Generator~~]		In [[number theory]], '''Mills' constant''' is defined as the smallest positive [[real number]] ''A'' such that the [[floor function\|floor]] of the [[double exponential function]]

			:<math> \lfloor A^{3^{n}} \rfloor</math>

			is a [[prime number]], for all positive integers ''n''. This constant is named after [[William H. Mills]] who proved in 1947 the existence of ''A'' based on results of [[Guido Hoheisel]] and [[Albert Ingham]] on the [[prime gap]]s.
			Its value is unknown, but if the [[Riemann hypothesis]] is true it is approximately 1.3063778838630806904686144926... {{OEIS\|A051021}}.

			==Mills primes==
			The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins
			:2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, ... {{OEIS\|A051254}}.

			If ''a<sub>i</sub>'' denotes the ''i''<sup>th</sup> prime in this sequence, then ''a<sub>i</sub>'' can be calculated as the smallest prime number larger than <math>a_{i-1}^3</math>. In order to ensure that rounding <math>A^{3^n}</math>, for ''n'' = 1, 2, 3, …, produces this sequence of primes, it must be the case that <math>a_i < (a_{i-1}+1)^3</math>. The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large [[cubic number]]s, which is sufficient to prove this inequality if we start from a sufficiently large first prime <math>a_1</math>. The Riemann hypothesis implies that there exists a prime between any two consecutive cubes, allowing the ''sufficiently large'' condition to be removed, and allowing the sequence of Mills' primes to begin at ''a''<sub>1</sub> = 2.

			Currently, the largest known Mills prime (under the Riemann hypothesis) is
			:<math>\displaystyle (((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220,</math>
			which is 20,562 digits long.

			==Numerical calculation==
			By calculating the sequence of Mills primes, one can approximate Mills' constant as
			:<math>A\approx a(n)^{1/3^n}.</math>
			{{harvtxt\|Caldwell\|Cheng\|2005}} used this method to compute almost seven thousand base 10 digits of Mills' constant under the assumption that the [[Riemann hypothesis]] is true. There is no closed-form formula known for Mills' constant, and it is not even known whether this number is [[rational number\|rational]] {{harv\|Finch\|2003}}.

			==See also==
			*[[Formula for primes]]

			==References==
			*{{citation\|last1=Caldwell\|first1=Chris K.\|last2=Cheng\|first2=Yuanyou\|title=Determining Mills' Constant and a Note on Honaker's Problem\|journal=Journal of Integer Sequences\|volume=8\|year=2005\|issue=5.4.1\|url=http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html}}.

			*{{citation\|first=Steven R.\|last=Finch\|title=Mathematical Constants\|year=2003\|publisher=Cambridge University Press\|isbn=0-521-81805-2\|contribution=Mills' Constant\|pages=130–133\|url=ftp://s208.math.msu.su/469000/dbcd69f8d83a96354dd49d21572c6432}}.

			*{{citation\|first=W. H.\|last=Mills\|title=A prime-representing function\|journal=[[Bulletin of the American Mathematical Society]]\|volume=53\|year=1947\|page=604\|doi=10.1090/S0002-9904-1947-08849-2\|issue=6\|url = http://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf}}.

			==External links==
			* {{MathWorld\|urlname=MillsConstant\|title=Mills' Constant}}
			* [http://blogs.ethz.ch/kowalski/2009/04/02/who-remembers-the-mills-number/ Who remembers the Mills number?], E. Kowalski.
			* [http://www.youtube.com/watch?v=6ltrPVPEwfo Awesome Prime Number Constant], Numberphile.

			{{Prime number classes}}

			[[Category:Mathematical constants]]
			[[Category:Prime numbers]]

en>Copyeditor42: fix pressure formula for kPa units as per text (was in mbar)

2012-07-24T21:48:49Z

fix pressure formula for kPa units as per text (was in mbar)

New page

Hi! <br>My name is Alex and I'm a 22 years old boy from Netherlands.<br><br>Feel free to visit my webpage [http://Horochicago.com/WP/?attachment_id=62 Fifa coin Generator]