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, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 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).