Radical of an integer
In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product:
The radical plays a central role in the statement of the abc conjecture.Examples
Radical numbers for the first few positive integers are
For example,
and thereforeProperties
The function is multiplicative.
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. There is no known polynomial-time algorithm for computing the square-free part of an integer.
The definition is generalized to the largest t-free divisor of n,, which are multiplicative functions which act on prime powers as
The cases t=3 and t=4 are tabulated in and.
The notion of the 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,
For any integer, the nilpotent elements of the finite ring are all of the multiples of.
