# Square-free

(Redirected from Square free)

In mathematics, a square-free integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32. The small 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, ... (sequence A005117 in OEIS)

## Equivalent characterizations of square-free numbers

The integer n is square-free if and only if in the prime factorization of n, no prime number occurs more than once. Another way of stating the same is that for every prime divisor p of n, the prime p does not 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.

The positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function.

The positive integer n is square-free if and only if all abelian groups of order n are isomorphic, which is the case if and only if all of them are cyclic. This follows from the classification of finitely generated abelian groups.

The integer n is square-free iff the factor ring Z / nZ (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ 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 divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if n is square-free.

Given the positive integer n, define the radical of the integer n by

equal to the product of the prime numbers p dividing n. Then the square-free n are exactly the solutions of n = rad(n).

## Distribution of square-free numbers

If Q(x) denotes the number of square-free integers between 1 and x, then

[itex]Q(x) = \frac{6x}{\pi^2} + O(\sqrt{x})[itex]

(see pi and big O notation). The asymptotic/natural density of square-free numbers is therefore

[itex]\lim_{x\to\infty} \frac{Q(x)}{x} = \frac{6}{\pi^2} = \frac{1}{\zeta(2)}[itex]

where ζ is the Riemann zeta function.

Likewise, if Q(x,n) denotes the number of nth power-free integers between 1 and x, one can show

• Art and Cultures
• Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
• Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
• Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
• United States (http://www.academickids.com/encyclopedia/index.php/United_States)
• World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
• Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
• Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
• Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
• Space and Astronomy
• Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)