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In mathematics, a square-free integer n is one divisible by no perfect square, except 1. Equivalently, n is square-free 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. For example, 10 is square-free but 20 is not.
The integer n is square-free if 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.
The positive integer n is square-free iff μ(n) ≠ 0, where μ denotes the Möbius function.
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: a <= b iff a divides b. This partially ordered set is always a distributive lattice. It is a boolean algebra if and only if n is square-free.
If Q(x) denotes the number of square-free numbers less than or equal to x, then
Equivalent characterizations of square-free numbers
Distribution of square-free numbers
(see pi and big O notation). The density of square-free numbers is therefore
