Water rocket - Revision history

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2015-01-09T19:37:28Z

Added {{ref improve}} tag to article (TW)

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en>Super48paul: Reverted 2 edits by 202.156.10.13 identified as test/vandalism using STiki {{subst:uw-vandalism#}}

2014-03-04T08:19:19Z

Reverted 2 edits by 202.156.10.13 identified as test/vandalism using STiki {{subst:uw-vandalism#}}

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	In mathematics, a '''noncototient''' is a positive integer ''n'' that cannot be expressed as the difference between a positive integer ''m'' and the number of [[coprime]] integers below it. That is, ''m'' − φ(''m'') = ''n'', where φ stands for [[Euler's totient function]], has no solution for ''m''. The ''[[cototient]]'' of ''n'' is defined as ''n'' − φ(''n''), so a '''noncototient''' is a number that is never a cototient.

	It is conjectured that all noncototients are even. This follows from a modified form of the [[Goldbach conjecture]]: if the even number ''n'' can be represented as a sum of two distinct primes ''p'' and ''q,'' then

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	~~pq - \varphi(pq)~~ = ~~pq - (p-1)(q-1)~~ = p~~+q-1~~ = n-1. \,
	<~~/math~~>

	~~It is expected that every even number larger than 6 is~~ a ~~sum~~ of ~~distinct primes, so probably no odd number larger than 5~~ is ~~a noncototient~~. ~~The remaining odd numbers are covered by~~ the ~~observations~~ <~~math~~>~~1=2-\phi(2), 3 = 9 - \phi(9)~~<~~/math~~> and <~~math~~>~~5 = 25 - \phi(25)~~<~~/math~~>.

	~~The first few noncototients are:~~

	~~[[10 (number)\|10]]~~, ~~[[26 (number)\|26]]~~, ~~[[34 (number)\|34]]~~, ~~[[50 (number)\|50]]~~, ~~[[52 (number)\|52]]~~, ~~[[58~~ (~~number)\|58]]~~, ~~[[86 (number)\|86]]~~, ~~[[100 (number~~)~~\|100]], [[116~~ (number)\|116]], [[122 (number)\|122]], [[130 (number)\|130]], [[134 (number)\|134]], [[146 (number)\|146]], [[154 (number)\|154]], [[170 (number)\|170]], [[172 (number)\|172]], 186, 202, 206, 218, [[222 (number)\|222]], 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, ~~412~~, ~~436, 466, 470, 474, 482, 490, 518, 520 {{OEIS\|id=A005278}}~~

	~~[[Paul Erdős\|Erdős]] (1913-1996)~~ and ~~[[Wacław Sierpiński\|Sierpinski]] (1882-1969~~) ~~asked whether there exist infinitely many noncototients~~. ~~This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family~~ <~~math~~> ~~2^k \cdot 509203~~<~~/math~~> ~~is an example. Since then other infinite families,~~ of ~~roughly~~ the ~~same form, have been given by Flammenkamp~~ and ~~Luca (2000).~~

	~~==References==~~
	* {{cite journal \| zbl=0820.11003 \| last1=Browkin \| first1=J. \| last2=Schinzel \| first2=A. \| title=On integers not of the ~~form n~~-~~φ(n) \| journal=Colloq. Math. \| volume=68 \| number=1 \| pages=55–58 \| year=1995 }}~~
	* {{cite journal \| zbl=0965.11003 \| last1=Flammenkamp \| first1=A. \| last2=Luca \| first2=F. \| title=Infinite families of ~~noncototients \| journal=Colloq~~. ~~Math. \| volume=86 \| number=1 \| pages=37–41 \| year=2000 }}~~
	* {{cite book \|last=Guy \| first=Richard K. \| authorlink=Richard K. Guy \| title=Unsolved problems in number theory \| publisher=[[Springer-Verlag]] \|edition=3rd \| year=2004 \|isbn=978-0-387-20860-2 \| zbl=1058.11001 \| pages=138–142}}

	~~== External links ==~~
	* [http://mathworld.wolfram.com/Noncototient.html Noncototient definition from MathWorld]

	~~{{Totient}}~~
	~~{{Classes~~ of ~~natural numbers}}~~

	~~[[Category:Integer sequences]]~~

en>The Anome: /* External links */ Category:Science demonstrations

2014-01-15T19:09:25Z

External links: Category:Science demonstrations

← Older revision		Revision as of 21:09, 15 January 2014
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	~~Contented to meet you! My husband~~ and ~~my name is Eusebio in addition , I think it to become quite good when everyone say~~ it. ~~As a man what While i really like~~ is ~~coming off as but I~~'m ~~thinking towards starting something new~~. ~~I work~~ as ~~an get clerk~~. ~~My house~~ is ~~this in~~ [~~http~~:~~//www.wikipedia.org/wiki/South+Carolina South Carolina] yet I don~~'~~t plan onto [http~~:/~~/En~~.~~Wiktionary.org~~/~~wiki~~/~~changing changing] it~~. ~~You will be able to find my website here~~: ~~http://prometeu~~.~~net~~<br><br>~~Feel free to visit my web page:~~ [http://~~prometeu~~.~~net gem hack clash~~ of ~~clans~~]		In mathematics, a '''noncototient''' is a positive integer ''n'' that cannot be expressed as the difference between a positive integer ''m'' and the number of [[coprime]] integers below it. That is, ''m'' − φ(''m'') = ''n'', where φ stands for [[Euler's totient function]], has no solution for ''m''. The ''[[cototient]]'' of ''n'' is defined as ''n'' − φ(''n''), so a '''noncototient''' is a number that is never a cototient.

			It is conjectured that all noncototients are even. This follows from a modified form of the [[Goldbach conjecture]]: if the even number ''n'' can be represented as a sum of two distinct primes ''p'' and ''q,'' then

			: <math>
			pq - \varphi(pq) = pq - (p-1)(q-1) = p+q-1 = n-1. \,
			</math>

			It is expected that every even number larger than 6 is a sum of distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations <math>1=2-\phi(2), 3 = 9 - \phi(9)</math> and <math>5 = 25 - \phi(25)</math>.

			The first few noncototients are:

			[[10 (number)\|10]], [[26 (number)\|26]], [[34 (number)\|34]], [[50 (number)\|50]], [[52 (number)\|52]], [[58 (number)\|58]], [[86 (number)\|86]], [[100 (number)\|100]], [[116 (number)\|116]], [[122 (number)\|122]], [[130 (number)\|130]], [[134 (number)\|134]], [[146 (number)\|146]], [[154 (number)\|154]], [[170 (number)\|170]], [[172 (number)\|172]], 186, 202, 206, 218, [[222 (number)\|222]], 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, 518, 520 {{OEIS\|id=A005278}}

			[[Paul Erdős\|Erdős]] (1913-1996) and [[Wacław Sierpiński\|Sierpinski]] (1882-1969) asked whether there exist infinitely many noncototients. This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family <math> 2^k \cdot 509203</math> is an example. Since then other infinite families, of roughly the same form, have been given by Flammenkamp and Luca (2000).

			==References==
			* {{cite journal \| zbl=0820.11003 \| last1=Browkin \| first1=J. \| last2=Schinzel \| first2=A. \| title=On integers not of the form n-φ(n) \| journal=Colloq. Math. \| volume=68 \| number=1 \| pages=55–58 \| year=1995 }}
			* {{cite journal \| zbl=0965.11003 \| last1=Flammenkamp \| first1=A. \| last2=Luca \| first2=F. \| title=Infinite families of noncototients \| journal=Colloq. Math. \| volume=86 \| number=1 \| pages=37–41 \| year=2000 }}
			* {{cite book \|last=Guy \| first=Richard K. \| authorlink=Richard K. Guy \| title=Unsolved problems in number theory \| publisher=[[Springer-Verlag]] \|edition=3rd \| year=2004 \|isbn=978-0-387-20860-2 \| zbl=1058.11001 \| pages=138–142}}

			== External links ==
			* [http://mathworld.wolfram.com/Noncototient.html Noncototient definition from MathWorld]

			{{Totient}}
			{{Classes of natural numbers}}

			[[Category:Integer sequences]]

61.197.222.158: /* Predicting Peak Height */

2012-08-09T07:04:32Z

Predicting Peak Height

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