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| In [[number theory]], the '''Erdős–Kac theorem''', named after [[Paul Erdős]] and [[Mark Kac]], and also known as the fundamental theorem of [[probabilistic number theory]], states that if ω(''n'') is the number of distinct [[prime factor]]s of ''n'', then, loosely speaking, the [[probability distribution]] of
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| : <math> \frac{\omega(n) - \log\log n}{\sqrt{\log\log n}} </math> | |
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| is the standard [[normal distribution]]. This is a deep extension of the [[Hardy–Ramanujan theorem]], which states that the [[Normal order of an arithmetic function|normal order]] of ω(''n'') is log log ''n'' with a typical error of size <math>\sqrt{\log\log n}</math>.
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| More precisely, for any fixed ''a'' < ''b'',
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| :<math>\lim_{x \rightarrow \infty} \left ( \frac {1}{x} \cdot \#\left\{ n \leq x : a \le \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}} \le b \right\} \right ) = \Phi(a,b) </math>
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| where <math>\Phi(a,b)</math> is the normal (or "Gaussian") distribution, defined as
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| : <math>\Phi(a,b)= \frac{1}{\sqrt{2\pi}}\int_a^b e^{-t^2/2} \, dt. </math> | |
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| Stated somewhat heuristically, what Erdős and Kac proved was that if ''n'' is a randomly chosen large integer, then the number of distinct prime factors of ''n'' has approximately the normal distribution with mean and variance log log ''n''.
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| This means that the construction of a number around one billion requires on average three primes.
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| For example 1,000,000,003 = 23 × 307 × 141623.
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| {| class="wikitable" border="2" style="text-align:center"
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| |-
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| ! ''n''
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| !Number of
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| Digits in ''n''
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| ! Average number
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| of distinct primes
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| ! standard
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| deviation
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| |-
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| |1,000
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| |4
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| |2
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| |1.4
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| |-
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| |1,000,000,000
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| |10
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| |3
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| |1.7
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| |-
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| |1,000,000,000,000,000,000,000,000
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| |25
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| |4
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| |2
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| |-
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| |10<sup>65</sup>
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| |66
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| |5
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| |2.2
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| |-
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| |10<sup>9,566</sup>
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| |9,567
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| |10
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| |3.2
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| |-
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| |10<sup>210,704,568</sup>
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| |210,704,569
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| |20
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| |4.5
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| |-
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| |10<sup>10<sup>22</sup></sup>
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| |10<sup>22</sup>+1
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| |50
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| |7.1
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| |-
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| |10<sup>10<sup>44</sup></sup>
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| |10<sup>44</sup>+1
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| |100
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| |10
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| |-
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| |10<sup>10<sup>434</sup></sup>
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| |10<sup>434</sup>+1
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| |1000
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| |31.6
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| |}
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| [[Image:EKT plot.svg|thumb|300px|right|A spreading Gaussian distribution of distinct primes illustrating the Erdos-Kac theorem]]
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| Around 12.6% of 10,000 digit numbers are constructed from 10 distinct prime numbers and around 68% (±σ) are constructed from between 7 and 13 primes.
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| A hollow sphere the size of the planet Earth filled with fine sand would have around 10<sup>33</sup> grains. A volume the size of the observable universe would have around 10<sup>93</sup> grains of sand. There might be room for 10<sup>185</sup> quantum strings in such a universe.
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| Numbers of this magnitude—with 186 digits—would require on average only 6 primes for construction.
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| ==References==
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| * {{cite journal | last1=Erdős | first1=Paul | author1-link=Paul Erdős | last2=Kac | first2=Mark | author2-link=Mark Kac | title=The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions | journal=[[American Journal of Mathematics]] | volume=62 | number=1/4 | year=1940 | pages=738–742 | zbl=0024.10203 | issn=0002-9327 }}
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| * {{cite book | last1=Kuo | first1=Wentang | last2=Liu | first2=Yu-Ru | chapter=The Erdős–Kac theorem and its generalizations | pages=209-216 | editor1-last=De Koninck | editor1-first=Jean-Marie | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 | location=Providence, RI | publisher=[[American Mathematical Society]] | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1187.11024 }}
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| * {{cite book | last=Kac | first=Mark | title=Statistical Independence in Probability, Analysis and Number Theory |year=1959 | publisher=John Wiley and Sons, Inc. }}
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| ==External links==
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| * {{MathWorld|urlname=Erdos-KacTheorem|title=Erdős–Kac Theorem}}
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| * [http://www.youtube.com/watch?v=4ivoaFLQ4vM#generator Timothy Gowers: The Importance of Mathematics (part 6, 4 mins in) and (part 7)]
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| {{DEFAULTSORT:Erdos-Kac theorem}}
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| [[Category:Paul Erdős|Kac theorem]]
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| [[Category:Normal distribution]]
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| [[Category:Theorems about prime numbers]]
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