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 the product mentioned):

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

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,

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

and therefore

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

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.^[1] 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  A007948 and  A058035.

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 K_ε such that, for all triples of coprime positive integers a, b, and c satisfying a + b = c,

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

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

See also

• Radical of an ideal

References

1. (sequence A007947 in the OEIS)

• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. p. 102. ISBN 0-387-20860-7.