View source for Simula

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

: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}}

==Equivalent characterizations==
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&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.

The positive integer ''n'' is square-free [[if and only if]] &mu;(''n'')&nbsp;≠&nbsp;0, where μ denotes the [[Möbius function]].

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.

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.

The [[radical of an integer]] is always square-free: an integer is square-free if it is equal to its radical.

==Dirichlet generating function==
The [[Dirichlet generating function]] for the square-free numbers is

:<math> \frac{\zeta(s)}{\zeta(2s) } = \sum_{n=1}^{\infty}\frac{ |\mu(n)|}{n^{s}} </math> where &zeta;(''s'') is the [[Riemann zeta function]].

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

==Distribution==
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:

:<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>

:<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>

This argument can be made rigorous, and a very elementary estimate yields

:<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]]). 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
by [[Arnold Walfisz]]<ref>A. Walfisz. "Weylsche Exponentialsummen in der neueren Zahlentheorie" (VEB deutscher Verlag der Wissenschaften, Berlin 1963.</ref> and we have
:<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>
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
:<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:

{{OEIS2C|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 factorization of every number is unique, so also 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 in 1985 for all sufficiently large integers by [[András Sárközy]],<ref>András Sárközy. On divisors of binomial coefficients, I. 
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>

==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 generating function]]s 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=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 }}
* {{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 }}

{{Divisor classes}}

{{Use dmy dates|date=September 2010}}

{{DEFAULTSORT:Square-Free Integer}}
[[Category:Number theory]]
[[Category:Integer sequences]]