Radical of an integer

In number theory, the radical of a positive integer n is defined as the product of the prime numbers dividing n:

$\displaystyle {\mbox{rad}}(n)=\prod _{\scriptstyle p\mid n \atop p~{\rm {prime}}}p.\,$

For example,

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

and therefore

${\mbox{rad}}(504)=2\cdot 3\cdot 7=42.\,$

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.

Radical numbers for the first few positive integers are 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (sequence A007947 in the OEIS).

The function ${\mbox{rad}}$ is multiplicative.

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 Z/nZ are all of the multiples of rad(n).

References

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