A square-free number is a natural number that contains no powers greater than 1 in its prime factorization. In other words, if $x$ is our number, and

$$ x = \prod_{i=1}^r p_i^{a_i} $$

is the prime factorization of $x$ into $r$ distinct primes, then $a_i \ge 2$ is always false for square-free $x$ .

Note: we assume here that $x$ itself must be greater than 1; hence 1 is not considered square-free. However, one must be alert to the particular context in which ``square-free'' is used as to whether this is considered the case.

The name derives from the fact that if any $a_i$ were to be greater than or equal to two, we could be sure that at least one square divides $x$ (namely, $p_i^2$ .)

Asymptotic Analysis

The asymptotic density of square-free numbers is $\frac{6}{\pi^2}$ which can be proved by application of a square-free variation of the sieve of Eratosthenes as follows:

	is squarefree

References

1

R. C. Baker and J. Pintz.
The distribution of square-free numbers.
Acta Arith., 46:73-79, 1985.
Zbl 0535.10045.