Square-free integer
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In mathematics, a square-free, or quadratfrei, 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
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
- m = rad(n),
equal to the product of the prime numbers p dividing n. This is also called the square-free part of the integer 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
- <math>Q(x) = \frac{6x}{\pi^2} + O(\sqrt{x})</math>
(see pi and big O notation). 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.
Likewise, if Q(x,n) denotes the number of nth power-free integers between 1 and x, one can show
- <math>\lim_{x\to\infty} \frac{Q(x,n)}{x} = \frac{1}{\zeta(n)}.</math>
Erdös 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.de:Quadratfrei es:Libre de cuadrados fr:Sans carré gl:Libre de cadrados it:Intero libero da quadrati nl:Kwadraatvrij sl:Deljivost brez kvadrata zh:無平方数因数的数