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, ...
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. 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
(see pi and big O notation). The asymptotic/natural density of square-free numbers is therefore
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
Last updated: 08-24-2005 11:38:54
