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:

$\displaystyle\mathrm(n)=\prod_p$

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
$\operatorname(504) = 2 \cdot 3 \cdot 7 = 42$

Properties

The function $\mathrm$ 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 $n$, $\mathrm_t$, which are multiplicative functions which act on prime powers as

$\mathrm_t(p^e) = p^$

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 $a+b=c$,

$c < K_\varepsilon\, \operatorname(abc)^$

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

References

{{reflist

Multiplicative functions

de:Zahlentheoretische Funktion#Multiplikative Funktionen