Square-free integer

A natural number is called square-free if there is no one except the square number that divides this number. In other words, occurs in the unique prime factorization of a square-free number to not prime more than once.

For example, the number is 6 = 2.3 square-free, while 54 = 2.32.3 is not square-free. The first 20 square-free numbers are

Properties

The Möbius function on the site is accurate then equal to 0 if it is square-free.

A number is square-free if and only if the residue class ring is reduced, that is, except when the zero no nilpotent element is contained.

The asymptotic probability that a randomly chosen number is square-free, is the Riemann ζ - function is. This means that the probability that a uniformly distributed from selected natural number is square-free, converges to.

General definition

A different from 0 element of a factorial ring is called square-free if, in its up to order and multiplication by units of the ring unique prime factorization (where a unit of the ring ) are all different from zero exponent equal to 1.

It should be and the formal derivation, then is square-free if it is. Thus, it is always square-free for any polynomial.

• Number Theory