Square-free Integer - Equivalent Characterizations

Equivalent Characterizations

The positive 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 factor 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. An immediate result of this definition is that all prime numbers are square-free.

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 if and only 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.

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.

The radical of an integer is always square-free: an integer is square-free if it is equal to its radical.