Squarefree

Squarefree is a term used in number theory and algebra. In number theory, a squarefree positive integer is one that is not divisible by the square of any prime. Equivalently, in prime factorization n = p1 p2 ... pk, all primes are distinct. The number 1 is usually considered squarefree. For example, 30 = 2 · 3 · 5 is squarefree, but 12 = 2^2 · 3 is not.

The radical of n, rad(n), is the product of the distinct prime divisors of n; for squarefree n this radical equals n. The Möbius function μ(n) is zero if n has a squared prime factor and otherwise μ(n) = (−1)^k where k is the number of prime factors; thus μ(n) ≠ 0 exactly for squarefree n. The proportion of squarefree numbers up to x is asymptotic to 6/π^2, and the counting function Q(x) ~ 6x/π^2.

Squarefree numbers can be generated by sieve methods that mark multiples of p^2 to exclude non-squarefree candidates. They play a role in multiplicative number theory, including in formulas that involve the Möbius function and in probabilistic models of random integers.

In algebra, a non-constant polynomial f in a field F[x] is squarefree if it is not divisible by the square of any non-constant polynomial; equivalently gcd(f, f′) = 1 when the characteristic of F does not cause the derivative to vanish. If f′ = 0 (as in inseparable polynomials in characteristic p), f is not squarefree. Squarefreeness of polynomials is important in polynomial factorization, the study of discriminants, and the structure of field extensions.