Square-free integer

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, p does not divide n / p.

The integer n ≠ 1 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.

A natural number n is square-free iff μ(n) ≠ 0, where μ denotes the Möbius function.