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74.136.34.201: /* Construction */

2014-09-17T01:44:41Z

Construction

← Older revision		Revision as of 03:44, 17 September 2014
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	~~In [[number theory]],~~ the '~~''radical''' of a [[positive number\|positive]] [[integer]] ''n'' is defined as the product of the [[prime number]]~~s ~~dividing ''n'':~~		The name of the writer is Numbers but it's not the most masucline title out there. Doing ceramics is what adore doing. Hiring is her day job now but she's usually wanted her personal business. California is exactly where I've always been residing and I adore every working day residing right here.<br><br>Here is my page - [http://www.dinnerfor6.com/home/profile/DLangridg http://www.dinnerfor6.com]

	~~:<math>\displaystyle\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ prime}}p</math>~~

	~~== Examples ==~~

	~~Radical numbers for~~ the ~~first few positive integers are~~
	~~: [[1 (number)\|1]], [[2 (number)\|2]], [[3 (number)\|3]], 2, [[5 (number)\|5]], [[6 (number)\|6]], [[7 (number)\|7]], 2, 3, [[10 (number)\|10]],~~ ... ~~{{OEIS\|A007947}}~~.

	~~For example,~~
	:<~~math~~>~~504 = 2^3 \cdot 3^2 \cdot 7~~<~~/math~~>

	~~and therefore~~
	:~~<math>\mathrm{rad}(504) = 2 \cdot 3 \cdot 7 = 42<~~/~~math>~~

	~~== Properties ==~~

	~~The function <math>\mathrm{rad}<~~/~~math> is [[multiplicative function\|multiplicative]] (but not [[Completely multiplicative function\|completely multiplicative]])~~.

	~~The radical of any integer ''n'' is the largest [[square-free integer\|square-free]] divisor of ''n''~~. ~~The definition is generalized to the largest ''t''-free divisor of ''n'', <math>\mathrm{rad}_t<~~/~~math>, which are multiplicative functions which act on prime powers as~~
	~~: <math>\mathrm{rad}_t(p^e) = p^{\mathrm{min}(e, t - 1)}<~~/~~math>~~

	~~The cases ''t''=3 and ''t''=4 are tabulated in {{OEIS2C\|A007948}} and {{OEIS2C\|A058035}}.~~

	~~One of the most striking applications of the notion of radical occurs in the [[abc conjecture]], which states that, for any ''ε'' > 0, there exists a finite ''K<sub>ε<~~/~~sub>'' such that, for all triples of [[coprime]] positive integers ''a'', ''b'', and ''c'' satisfying ''a'' + ''b'' = ''c'',~~

	:~~<math>c < K_\varepsilon\, \operatorname{rad}(abc)^{1 + \varepsilon}</math>~~

	~~Furthermore, it can be shown that the [[nilpotent]] elements of <math>\mathbb{Z}~~/~~n\mathbb{Z}<~~/~~math> are all of the multiples of rad(''n'')~~.

	~~==See also==~~
	* [[Radical of an ideal]]

	~~==References==~~
	* {{cite book\|last=Guy\|first=Richard K.\|authorlink=Richard K. ~~Guy\|title=Unsolved Problems in Number Theory\|publisher=[[Springer-Verlag]]\|date=2004\|page=102\|isbn=0-387-20860-7}}~~

	~~[[Category:Number theory]]~~
	~~[[Category:Multiplicative functions]]~~
	~~{{numtheory-stub}}~~

	~~[[de:Zahlentheoretische Funktion#Multiplikative Funktionen]~~]

en>Igodard: /* Purpose */

2013-12-08T06:34:08Z

Purpose

← Older revision		Revision as of 08:34, 8 December 2013
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	~~I woke up yesterday and realized - I have been single for~~ [~~http://www.netpaw.org tour luke bryan~~] a ~~while at~~ the ~~moment tickets to luke bryan (~~[~~http~~:/~~/lukebryantickets.citizenswebcasting.com http~~:~~//lukebryantickets.citizenswebcasting.com~~]) ~~and~~ [~~http://lukebryantickets.asiapak.net luke bryan tour~~] ~~after much~~ [~~http://Www~~.~~Wikipedia~~.~~org/wiki/bullying bullying] from pals I today find myself opted for on line dating~~. ~~They promised me that there are plenty of pleasant~~, ~~standard~~ and ~~entertaining visitors to meet,~~ therefore ~~here goes the toss!~~<br>~~I make an effort to stay as toned as [http:~~//~~www~~.~~thefreedictionary~~.~~com/potential+coming potential coming]~~ to the gymnasium many times weekly. I appreciate my athletics and strive to play or watch as numerous a possible. I will frequently at Hawthorn suits being wintertime. Notice: If you really contemplated purchasing an activity I really don't ~~brain, I~~'~~ve observed the carnage~~ of ~~wrestling fits at stocktake sales.~~<br>~~My friends and household~~ are ~~awe~~-~~inspiring~~ and ~~hanging out with them at bar gigs or dinners is consistently a necessity. As I see that you can not own a nice dialogue with the sound I haven~~'t ~~ever been~~ in ~~to clubs~~. ~~I also have 2 quite cute and definitely cheeky puppies~~ that ~~are constantly ready to meet up fresh folks.~~<br><br>~~My blog post~~; ~~Luke Bryan Vip Tickets~~, [~~http:~~//~~lukebryantickets~~.~~hamedanshahr~~.~~com Http~~:~~//Lukebryantickets.Hamedanshahr.Com~~],		In [[number theory]], the '''radical''' of a [[positive number\|positive]] [[integer]] ''n'' is defined as the product of the [[prime number]]s dividing ''n'':

			:<math>\displaystyle\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ prime}}p</math>

			== Examples ==

			Radical numbers for the first few positive integers are
			: [[1 (number)\|1]], [[2 (number)\|2]], [[3 (number)\|3]], 2, [[5 (number)\|5]], [[6 (number)\|6]], [[7 (number)\|7]], 2, 3, [[10 (number)\|10]], ... {{OEIS\|A007947}}.

			For example,
			:<math>504 = 2^3 \cdot 3^2 \cdot 7</math>

			and therefore
			:<math>\mathrm{rad}(504) = 2 \cdot 3 \cdot 7 = 42</math>

			== Properties ==

			The function <math>\mathrm{rad}</math> is [[multiplicative function\|multiplicative]] (but not [[Completely multiplicative function\|completely multiplicative]]).

			The radical of any integer ''n'' is the largest [[square-free integer\|square-free]] divisor of ''n''. The definition is generalized to the largest ''t''-free divisor of ''n'', <math>\mathrm{rad}_t</math>, which are multiplicative functions which act on prime powers as
			: <math>\mathrm{rad}_t(p^e) = p^{\mathrm{min}(e, t - 1)}</math>

			The cases ''t''=3 and ''t''=4 are tabulated in {{OEIS2C\|A007948}} and {{OEIS2C\|A058035}}.

			One of the most striking applications of the notion of radical occurs in the [[abc conjecture]], which states that, for any ''ε'' > 0, there exists a finite ''K<sub>ε</sub>'' such that, for all triples of [[coprime]] positive integers ''a'', ''b'', and ''c'' satisfying ''a'' + ''b'' = ''c'',

			:<math>c < K_\varepsilon\, \operatorname{rad}(abc)^{1 + \varepsilon}</math>

			Furthermore, it can be shown that the [[nilpotent]] elements of <math>\mathbb{Z}/n\mathbb{Z}</math> are all of the multiples of rad(''n'').

			==See also==
			* [[Radical of an ideal]]

			==References==
			* {{cite book\|last=Guy\|first=Richard K.\|authorlink=Richard K. Guy\|title=Unsolved Problems in Number Theory\|publisher=[[Springer-Verlag]]\|date=2004\|page=102\|isbn=0-387-20860-7}}

			[[Category:Number theory]]
			[[Category:Multiplicative functions]]
			{{numtheory-stub}}

			[[de:Zahlentheoretische Funktion#Multiplikative Funktionen]]

109.210.42.44: words direct and indirect were erroneously permuted in the sentence

2012-03-11T17:46:51Z

words direct and indirect were erroneously permuted in the sentence

New page

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