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In [[mathematics]], a '''square-free''', or '''quadratfrei''', [[integer]] is one [[divisor|divisible]] by no [[square number|perfect square]], except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 3<sup>2</sup>. The smallest positive square-free numbers are
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


:1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, ... {{OEIS|id=A005117}}
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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[[Ring theory]] generalizes the concept of being [[square-free]].
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==Equivalent characterizations==
'''MathML'''
The positive integer ''n'' is square-free if and only if in the [[canonical representation of a positive integer|prime factorization]] of ''n'', no [[prime number]] occurs more than once. Another way of stating the same is that for every prime [[divisor|factor]] ''p'' of ''n'', the prime ''p'' does not divide&nbsp;''n''&nbsp;/&nbsp;''p''. Yet another formulation: ''n'' is square-free if and only if in every factorization ''n''&nbsp;=&nbsp;''ab'', the factors ''a'' and ''b'' are [[coprime]].  An immediate result of this definition is that all prime numbers are square-free.
:<math forcemathmode="mathml">E=mc^2</math>


The positive integer ''n'' is square-free [[if and only if]] &mu;(''n'')&nbsp;≠&nbsp;0, where μ denotes the [[Möbius function]].
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


The [[Dirichlet series]] that generates the square-free numbers is
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


:<math> \frac{\zeta(s)}{\zeta(2s) } = \sum_{n=1}^{\infty}\frac{ |\mu(n)|}{n^{s}} </math> where &zeta;(''s'') is the [[Riemann zeta function]].
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


This is easily seen from the [[Euler product]]
==Demos==
:<math>  \frac{\zeta(s)}{\zeta(2s) } =\prod_p \frac{(1-p^{-2s})}{(1-p^{-s})}=\prod_p (1+p^{-s}). </math>


The positive integer ''n'' is square-free if and only if all [[abelian group]]s of [[order (group theory)|order]] ''n'' are [[group isomorphism|isomorphic]], which is the case if and only if all of them are [[cyclic group|cyclic]]. This follows from the classification of [[finitely generated abelian group]]s.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


The integer ''n'' is square-free if and only if the [[factor ring]] '''Z'''&nbsp;/&nbsp;''n'''''Z''' (see [[modular arithmetic]]) is a [[product of rings|product]] of [[field (mathematics)|field]]s. This follows from the [[Chinese remainder theorem]] and the fact that a ring of the form '''Z'''&nbsp;/&nbsp;''k'''''Z''' is a field if and only if ''k'' is a prime.


For every positive integer ''n'', the set of all positive divisors of ''n'' becomes a [[partially ordered set]] if we use [[divisor|divisibility]] as the order relation. This partially ordered set is always a [[distributive lattice]]. It is a [[Boolean algebra (structure)|Boolean algebra]] if and only if ''n'' is square-free.
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** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


The [[radical of an integer]] is always square-free.
==Test pages ==


==Distribution==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
Let ''Q''(''x'') denote the number of square-free (quadratfrei) integers between 1 and ''x''. For large ''n'', 3/4 of the positive integers less than ''n'' are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Because these events are independent, we obtain the approximation:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


:<math>Q(x) \approx x\prod_{p\ \text{prime}} \left(1-\frac{1}{p^2}\right) = x\prod_{p\ \text{prime}} \frac{1}{(1-\frac{1}{p^2})^{-1}} </math>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>Q(x) \approx x\prod_{p\ \text{prime}} \frac{1}{1+\frac{1}{p^2}+\frac{1}{p^4}+\cdots} = \frac{x}{\sum_{k=1}^\infty \frac{1}{k^2}} = \frac{x}{\zeta(2)} </math>
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
This argument can be made rigorous to yield:
 
:<math>Q(x) = \frac{x}{\zeta(2)} + O\left(\sqrt{x}\right) = \frac{6x}{\pi^2} + O\left(\sqrt{x}\right)</math>
 
(see [[pi]] and [[big O notation]]). Under the [[Riemann hypothesis]], the error term can be reduced:<ref>Jia, Chao Hua. "The distribution of square-free numbers", ''Science in China Series A: Mathematics'' '''36''':2 (1993), pp. 154–169. Cited in Pappalardi 2003, [http://www.mat.uniroma3.it/users/pappa/papers/allahabad2003.pdf A Survey on ''k''-freeness]; also see Kaneenika Sinha, "[http://www.math.ualberta.ca/~kansinha/maxnrevfinal.pdf Average orders of certain arithmetical functions]", ''Journal of the Ramanujan Mathematical Society'' '''21''':3 (2006), pp. 267–277.</ref>
:<math>Q(x) = \frac{x}{\zeta(2)} + O\left(x^{17/54+\varepsilon}\right) = \frac{6x}{\pi^2} + O\left(x^{17/54+\varepsilon}\right).</math>
 
See the race between the number of square-free numbers up to ''n'' and round(''n''/&zeta;(2)) on the OEIS:
 
[http://www.research.att.com/~njas/sequences/A158819 A158819 – (Number of square-free numbers ≤&nbsp;''n'')&nbsp;minus&nbsp;round(''n''/&zeta;(2)).  ]
 
The asymptotic/[[natural density]] of square-free numbers is therefore
 
:<math>\lim_{x\to\infty} \frac{Q(x)}{x} = \frac{6}{\pi^2} = \frac{1}{\zeta(2)}</math>
 
where ζ is the [[Riemann zeta function]] and 1/ζ(2) is approximately 0.6079 (over 3/5 of the integers are square-free).
 
Likewise, if ''Q''(''x'',''n'') denotes the number of ''n''-free integers (e.g. 3-free integers being cube-free integers) between 1 and ''x'', one can show
:<math>Q(x,n) = \frac{x}{\sum_{k=1}^\infty \frac{1}{k^n}} + O\left(\sqrt[n]{x}\right) = \frac{x}{\zeta(n)} + O\left(\sqrt[n]{x}\right).</math>
 
==Encoding as binary numbers==
If we represent a square-free number as the infinite product:
 
:<math>\prod_{n=0}^\infty {p_{n+1}}^{a_n}, a_n \in \lbrace 0, 1 \rbrace,\text{ and }p_n\text{ is the }n\text{th prime}. </math>
 
then we may take those <math>a_n</math> and use them as bits in a binary number, i.e. with the encoding:
 
:<math>\sum_{n=0}^\infty {a_n}\cdot 2^n</math>
 
e.g. The square-free number 42 has factorisation 2&nbsp;&times;&nbsp;3&nbsp;&times;&nbsp;7, or as an infinite product: 2<sup>1</sup>&nbsp;·&nbsp;3<sup>1</sup> &nbsp;·&nbsp;5<sup>0</sup>&nbsp;·&nbsp;7<sup>1</sup>&nbsp;·&nbsp;11<sup>0</sup>&nbsp;·&nbsp;13<sup>0</sup>&nbsp;·&nbsp;...; Thus the number 42 may be encoded as the binary sequence <tt>...001011</tt> or 11 decimal. (Note that the binary digits are reversed from the ordering in the infinite product.)
 
Since the prime factorisation of every number is unique, so then is every binary encoding of the square-free integers.
 
The converse is also true. Since every positive integer has a unique binary representation it is possible to reverse this encoding so that they may be 'decoded' into a unique square-free integer.
 
Again, for example if we begin with the number 42, this time as simply a positive integer, we have its binary representation <tt>101010</tt>. This 'decodes' to become 2<sup>0</sup>&nbsp;·&nbsp;3<sup>1</sup>&nbsp;·&nbsp;5<sup>0</sup>&nbsp;·&nbsp;7<sup>1</sup>&nbsp;·&nbsp;11<sup>0</sup>&nbsp;·&nbsp;13<sup>1</sup> =&nbsp;3&nbsp;&times;&nbsp;7&nbsp;&times;&nbsp;13 =&nbsp;273.
 
Among other things, this implies that the set of all square-free integers has the same [[cardinality]] as the set of all integers. In turn that leads to the fact that the in-order encodings of the square-free integers are a permutation of the set of all integers.
 
See sequences [[OEIS:A048672|A048672]] and [[OEIS:A064273|A064273]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]
 
==Erdős squarefree conjecture==
The [[central binomial coefficient]]
 
<math>{2n \choose n}</math>
 
is never squarefree for ''n'' > 4. This was proven in 1996 by [[Olivier Ramaré]] and [[Andrew Granville]].
 
==Squarefree core==
The [[multiplicative function]] <math>\mathrm{core}_t(n)</math> is defined
to map positive integers ''n'' to ''t''-free numbers by reducing the
exponents in the prime power representation modulo ''t'':
: <math>\mathrm{core}_t(p^e) = p^{e\mod t}.</math>
The value set of <math>\mathrm{core}_2</math>, in particular, are the
square-free integers. Their [[Dirichlet series|Dirichlet generating functions]] are
: <math>\sum_{n\ge 1}\frac{\mathrm{core}_t(n)}{n^s}
= \frac{\zeta(ts)\zeta(s-1)}{\zeta(ts-t)}</math>.
 
[[OEIS]] representatives are {{OEIS2C|A007913}} (''t''=2), {{OEIS2C|A050985}} (''t''=3) and {{OEIS2C|A053165}} (''t''=4).
 
== Notes ==
<references/>
 
==References==
 
*{{cite journal
|first1=Andrew
|last1=Granville
|first2=Olivier
|last2=Ramare
|title=Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients
|mr=1401709
|year=1996
|journal=Mathematika
|volume=43
|pages=73–107
|doi=10.1112/S0025579300011608
}}
 
{{Divisor classes navbox}}
{{Use dmy dates|date=September 2010}}
 
{{DEFAULTSORT:Square-Free Integer}}
[[Category:Number theory]]
[[Category:Integer sequences]]
 
[[ar:عدد صحيح خال من المربعات]]
[[bg:Безквадратно число]]
[[cs:Bezčtvercové celé číslo]]
[[de:Quadratfrei]]
[[es:Entero libre de cuadrados]]
[[eo:Kvadrato-libera entjero]]
[[fr:Entier sans facteur carré]]
[[gl:Libre de cadrados]]
[[it:Intero privo di quadrati]]
[[hu:Négyzetmentes szám]]
[[nl:Kwadraatvrij geheel getal]]
[[pt:Inteiro sem fator quadrático]]
[[ru:Бесквадратное число]]
[[sl:Deljivost brez kvadrata]]
[[zh:无平方数因数的数]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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