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:

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

The radical plays a central role in the statement of the abc conjecture.

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, ....

For example, $$504 = 2^3 \cdot 3^2 \cdot 7$$

and therefore $$\operatorname{rad}(504) = 2 \cdot 3 \cdot 7 = 42$$

Properties
The function $$\mathrm{rad}$$ 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 $$n$$. 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 $$n$$, $$\mathrm{rad}_t$$, which are multiplicative functions which act on prime powers as

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

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 $$\varepsilon > 0$$, there exists a finite $$K_\varepsilon$$ such that, for all triples of coprime positive integers $$a$$, $$b$$, and $$c$$ satisfying $$a+b=c$$,

$$c < K_\varepsilon\, \operatorname{rad}(abc)^{1 + \varepsilon}$$

For any integer $$n$$, the nilpotent elements of the finite ring $$\mathbb{Z}/n\mathbb{Z}$$ are all of the multiples of $$\operatorname{rad}(n)$$.

The Dirichlet series is


 * $$\prod_p \left(1+\frac{p^{1-s}}{1-p^{-s}}\right) = \sum_{n=1}^{\infty} \frac{\operatorname{rad}(n)}{n^s}$$