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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] |
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| :<math>\displaystyle\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ prime}}p</math>
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| == Examples ==
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| Radical numbers for the first few positive integers are
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| : [[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}}.
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| For example,
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| :<math>504 = 2^3 \cdot 3^2 \cdot 7</math>
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| and therefore
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| :<math>\mathrm{rad}(504) = 2 \cdot 3 \cdot 7 = 42</math> | |
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| == Properties ==
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| The function <math>\mathrm{rad}</math> is [[multiplicative function|multiplicative]] (but not [[Completely multiplicative function|completely multiplicative]]).
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| 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
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| : <math>\mathrm{rad}_t(p^e) = p^{\mathrm{min}(e, t - 1)}</math>
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| The cases ''t''=3 and ''t''=4 are tabulated in {{OEIS2C|A007948}} and {{OEIS2C|A058035}}.
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| 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'',
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| :<math>c < K_\varepsilon\, \operatorname{rad}(abc)^{1 + \varepsilon}</math> | |
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| 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'').
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| ==See also==
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| * [[Radical of an ideal]]
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| ==References==
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| * {{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}}
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| [[Category:Number theory]]
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| [[Category:Multiplicative functions]]
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| {{numtheory-stub}}
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| [[de:Zahlentheoretische Funktion#Multiplikative Funktionen]]
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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.
Here is my page - http://www.dinnerfor6.com