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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
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| :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}}
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| ==Equivalent characterizations==
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| 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 evenly divide ''n'' / ''p''. Yet another formulation: ''n'' is square-free if and only if in every factorization ''n'' = ''ab'', the factors ''a'' and ''b'' are [[coprime]]. An immediate result of this definition is that all prime numbers are square-free.
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| The positive integer ''n'' is square-free [[if and only if]] μ(''n'') ≠ 0, where μ denotes the [[Möbius function]].
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| 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.
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| The integer ''n'' is square-free if and only if the [[factor ring]] '''Z''' / ''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''' / ''k'''''Z''' is a field if and only if ''k'' is a prime.
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| 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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| The [[radical of an integer]] is always square-free: an integer is square-free if it is equal to its radical.
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| ==Dirichlet generating function==
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| The [[Dirichlet generating function]] for the square-free numbers is
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| :<math> \frac{\zeta(s)}{\zeta(2s) } = \sum_{n=1}^{\infty}\frac{ |\mu(n)|}{n^{s}} </math> where ζ(''s'') is the [[Riemann zeta function]]. | |
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| This is easily seen from the [[Euler product]]
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| :<math> \frac{\zeta(s)}{\zeta(2s) } =\prod_p \frac{(1-p^{-2s})}{(1-p^{-s})}=\prod_p (1+p^{-s}). </math>
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| ==Distribution==
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| 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:
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| :<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>
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| :<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>
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| This argument can be made rigorous, and a very elementary estimate yields
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| :<math>Q(x) = \frac{x}{\zeta(2)} + O\left(\sqrt{x}\right) = \frac{6x}{\pi^2} + O\left(\sqrt{x}\right)</math>
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| (see [[pi]] and [[big O notation]]). By exploiting the largest known zero-free region of the Riemann zeta function, due to [[Ivan Matveyevich Vinogradov]], [[:ru:Коробов, Николай Михайлович|M.N. Korobov]] and [[Hans-Egon Richert]], the maximal size of the error term has been reduced
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| by [[Arnold Walfisz]]<ref>A. Walfisz. "Weylsche Exponentialsummen in der neueren Zahlentheorie" (VEB deutscher Verlag der Wissenschaften, Berlin 1963.</ref> and we have
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| :<math>Q(x) = \frac{6x}{\pi^2} + O\left(x^{1/2}\exp\left(-c\frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right)\right).</math>
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| for some positive constant ''c''. Under the [[Riemann hypothesis]], the error term can be further 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> to yield
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| :<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>
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| See the race between the number of square-free numbers up to ''n'' and round(''n''/ζ(2)) on the OEIS:
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| {{OEIS2C|A158819}} – (Number of square-free numbers ≤ ''n'') minus round(''n''/ζ(2)). ]
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| The asymptotic/[[natural density]] of square-free numbers is therefore
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| :<math>\lim_{x\to\infty} \frac{Q(x)}{x} = \frac{6}{\pi^2} = \frac{1}{\zeta(2)}</math>
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| where ζ is the [[Riemann zeta function]] and 1/ζ(2) is approximately 0.6079 (over 3/5 of the integers are square-free).
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| 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
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| :<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>
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| ==Encoding as binary numbers==
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| If we represent a square-free number as the infinite product:
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| :<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>
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| then we may take those <math>a_n</math> and use them as bits in a binary number, i.e. with the encoding:
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| :<math>\sum_{n=0}^\infty {a_n}\cdot 2^n</math>
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| e.g. The square-free number 42 has factorisation 2 × 3 × 7, or as an infinite product: 2<sup>1</sup> · 3<sup>1</sup> · 5<sup>0</sup> · 7<sup>1</sup> · 11<sup>0</sup> · 13<sup>0</sup> · ...; 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.)
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| Since the prime factorization of every number is unique, so also is every binary encoding of the square-free integers.
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| 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. | |
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| 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> · 3<sup>1</sup> · 5<sup>0</sup> · 7<sup>1</sup> · 11<sup>0</sup> · 13<sup>1</sup> = 3 × 7 × 13 = 273.
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| 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.
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| See sequences [[OEIS:A048672|A048672]] and [[OEIS:A064273|A064273]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]
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| ==Erdős squarefree conjecture==
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| The [[central binomial coefficient]]
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| <math>{2n \choose n}</math>
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| is never squarefree for ''n'' > 4. This was proven in in 1985 for all sufficiently large integers by [[András Sárközy]],<ref>András Sárközy. On divisors of binomial coefficients, I.
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| J. Number Theory 20 (1985), no. 1, 70–80.</ref> and for all integers in 1996 by [[Olivier Ramaré]] and [[Andrew Granville]].<ref>Olivier Ramaré and Andrew Granville. Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika 43 (1996), no. 1, 73–107</ref>
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| ==Squarefree core==
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| The [[multiplicative function]] <math>\mathrm{core}_t(n)</math> is defined
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| to map positive integers ''n'' to ''t''-free numbers by reducing the
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| exponents in the prime power representation modulo ''t'':
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| : <math>\mathrm{core}_t(p^e) = p^{e\mod t}.</math>
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| The value set of <math>\mathrm{core}_2</math>, in particular, are the
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| square-free integers. Their [[Dirichlet generating function]]s are
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| : <math>\sum_{n\ge 1}\frac{\mathrm{core}_t(n)}{n^s} | |
| = \frac{\zeta(ts)\zeta(s-1)}{\zeta(ts-t)}</math>.
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| [[OEIS]] representatives are {{OEIS2C|A007913}} (''t''=2), {{OEIS2C|A050985}} (''t''=3) and {{OEIS2C|A053165}} (''t''=4).
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| == Notes ==
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| <references/>
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| == References ==
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| *{{cite journal | first1=Andrew | last1=Granville | first2=Olivier | last2=Ramaré | title=Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients | mr=1401709 | zbl=0868.11009 | year=1996 | journal=Mathematika | volume=43 | pages=73–107 | doi=10.1112/S0025579300011608 }}
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| * {{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=0-387-20860-7 | zbl=1058.11001 }}
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| {{Divisor classes}}
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Square-Free Integer}}
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| [[Category:Number theory]]
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| [[Category:Integer sequences]]
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