58.6.93.99: /* Applications and advantages */ Minor grammar fixes.

2014-10-16T06:12:56Z

Applications and advantages: Minor grammar fixes.

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	~~In [[number theory]], a '''totative''' of a given positive integer {{mvar\|n}}~~ is an ~~integer {{mvar\|k}} such that {{math\|0 < ''k'' < ''n''}} and {{mvar\|k}} is [[coprime]]~~ to~~ {{mvar\|n}}~~. ~~[[Euler's totient function]] φ(''n'') counts~~ the ~~number~~ of ~~totatives of ''n''~~. ~~The totatives under multiplication modulo ''n'' form~~ the ~~[[Multiplicative group of integers modulo n\|multiplicative group of integers modulo ''n'']]~~.		Carp fishing is definitely an awfully exciting activity though stipulations apply. Indeed there are many things you have to have as a way to enjoy your fishing session including having the right bait. A lot of the fun depends on trying to find those monstrously huge carp fish. These are very prominent in numerous places. You should use any sort of bait but boilies are among the best for all time. Through the years, these baits are becoming extremely popular. <br><br><br>

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	:<~~math~~> ~~0 < a_1 < a_2 \cdots < a_{\phi(n)} < n ,~~<~~/math~~>

	~~the mean square gap satisfies~~

	:<~~math~~> ~~\sum_{i=1}^{\phi(n)-1} (a_{i+1}-a_i)^2 < C n^2 / \phi(n)~~ <~~/math~~>

	for some ~~constant ''C''~~ and ~~this was proved by~~ [~~[Bob Vaughan]] and [[Hugh Montgomery (mathematician)\|Hugh Montgomery]~~].<~~ref~~>~~{{cite journal \| doi=10~~.~~2307/1971274 \| zbl=0591~~.~~10042 \| last1=Montgomery \| first1=H~~.L. ~~\| author1-link=Hugh Montgomery (mathematician) \| last2=Vaughan \| first2=R~~.C. ~~\| author2-link=Bob Vaughan \| title=On~~ the ~~distribution~~ of ~~reduced residues \| journal=Ann~~. ~~Math~~. ~~(2) \| volume=123 \| pages=311–333 \| year=1986 }}~~<~~/ref~~>

	~~==See also==~~
	*[~~[Reduced residue system]]~~

	~~==References==~~
	~~{{reflist}}~~
	* {{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 \| at=B40 }}~~

	~~==Further reading==~~
	*{{Citation \| last=Sándor \| first=Jozsef \| last2=Crstici \| first2=Borislav \| title=Handbook of number theory II \| location=Dordrecht \| publisher=Kluwer Academic \| year=2004 \| isbn=1-4020-~~2546~~-~~7 \| zbl=1079~~.~~11001 \| pages=242–250 }}~~

	~~==External links==~~
	*{{MathWorld \|title=Totative \|id=Totative}}
	*{{PlanetMath \|urlname=Totative \|title=totative}}

	~~[[Category:Modular arithmetic]]~~
	~~{{Numtheory-stub}}~~

98.28.166.53 at 20:35, 27 July 2012

2012-07-27T20:35:07Z

New page

In [[number theory]], a '''totative''' of a given positive integer {{mvar|n}} is an integer {{mvar|k}} such that {{math|0 < ''k'' < ''n''}} and {{mvar|k}} is [[coprime]] to {{mvar|n}}. [[Euler's totient function]] φ(''n'') counts the number of totatives of ''n''. The totatives under multiplication modulo ''n'' form the [[Multiplicative group of integers modulo n|multiplicative group of integers modulo ''n'']].

The distribution of totatives has been a subject of study. [[Paul Erdős]] conjectured that, writing the totatives of ''n'' as

:<math> 0 < a_1 < a_2 \cdots < a_{\phi(n)} < n ,</math>

the mean square gap satisfies

:<math> \sum_{i=1}^{\phi(n)-1} (a_{i+1}-a_i)^2 < C n^2 / \phi(n) </math>

for some constant ''C'' and this was proved by [[Bob Vaughan]] and [[Hugh Montgomery (mathematician)|Hugh Montgomery]].<ref>{{cite journal | doi=10.2307/1971274 | zbl=0591.10042 | last1=Montgomery | first1=H.L. | author1-link=Hugh Montgomery (mathematician) | last2=Vaughan | first2=R.C. | author2-link=Bob Vaughan | title=On the distribution of reduced residues | journal=Ann. Math. (2) | volume=123 | pages=311–333 | year=1986 }}</ref>

==See also==
*[[Reduced residue system]]

==References==
{{reflist}}
* {{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 | at=B40 }}

==Further reading==
*{{Citation | last=Sándor | first=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | zbl=1079.11001 | pages=242–250 }}

==External links==
*{{MathWorld |title=Totative |id=Totative}}
*{{PlanetMath |urlname=Totative |title=totative}}

[[Category:Modular arithmetic]]
{{Numtheory-stub}}

Semantic compression - Revision history

58.6.93.99: /* Applications and advantages */ Minor grammar fixes.

98.28.166.53 at 20:35, 27 July 2012