Square-free

In mathematics, an integer n is called square-free iff no perfect square except 1 divides n. Equivalently, n is square-free iff 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.

Equivalent characterizations of square-free numbers

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.

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 lattice. It is a boolean algebra if and only if n is square-free.

Distribution of square-free numbers

If Q(x) denotes the number of square-free numbers less than or equal to x, then

<math>Q(x) = \frac{6x}{\pi^2} + O(\sqrt{x})</math>

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

<math>\lim_{x\to\infty} \frac{Q(x)}{x} = \frac{6}{\pi^2}</math>