Radical of an integer
Radical numbers for the first few positive integers are
The radical of any integer n is the largest square-free divisor of n and so also described as the square-free kernel of n. The definition is generalized to the largest t-free divisor of n, , which are multiplicative functions which act on prime powers as
One of the most striking applications of the notion of radical occurs in the abc conjecture, which states that, for any ε > 0, there exists a finite Kε such that, for all triples of coprime positive integers a, b, and c satisfying a + b = c,
Furthermore, it can be shown that the nilpotent elements of are all of the multiples of rad(n).