Square-free integer

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In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32. The smallest square-free numbers are

Ring theory generalizes the concept of being square-free.


Equivalent characterizations of square-free numbers

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 Dirichlet series that generates the square-free numbers is

This is easily seen from the Euler product

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.

Distribution of square-free numbers

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