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 the product mentioned):

${\displaystyle \displaystyle \mathrm {rad} (n)=\prod _{\scriptstyle p\mid n \atop p{\text{ prime}}}p}$

## Examples

Radical numbers for the first few positive integers are

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (sequence A007947 in the OEIS).

For example,

${\displaystyle 504=2^{3}\cdot 3^{2}\cdot 7}$

and therefore

${\displaystyle \mathrm {rad} (504)=2\cdot 3\cdot 7=42}$

## Properties

The function ${\displaystyle \mathrm {rad} }$ is multiplicative (but not completely 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.[1] The definition is generalized to the largest t-free divisor of n, ${\displaystyle \mathrm {rad} _{t}}$, which are multiplicative functions which act on prime powers as

${\displaystyle \mathrm {rad} _{t}(p^{e})=p^{\mathrm {min} (e,t-1)}}$

The cases t=3 and t=4 are tabulated in and .

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 ab, and c satisfying a + b = c,

${\displaystyle c

Furthermore, it can be shown that the nilpotent elements of ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ are all of the multiples of rad(n).