Square-free

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 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 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.

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 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 numbers less than or equal to x, one can show

<math>\lim_{x\to\infty} \frac{Q(x,n)}{x} = \frac{1}{\zeta(n)}.<math>