thumb|The first thousand values of rad(n). rad(n) = n when n is [[Square-free integer|square-free.]]
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:
<math display=block>\displaystyle\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ primep</math>
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,
<math display=block>504 = 2^3 \cdot 3^2 \cdot 7</math>
and therefore
<math display=block>\operatorname{rad}(504) = 2 \cdot 3 \cdot 7 = 42</math>
Properties
The function <math>\mathrm{rad}</math> is multiplicative (but not completely multiplicative).
The radical of any integer <math>n</math> is the largest square-free divisor of <math>n</math> and so also described as the square-free kernel of <math>n</math>. There is no known polynomial-time algorithm for computing the square-free part of an integer.
The definition is generalized to the largest <math>t</math>-free divisor of <math>n</math>, <math>\mathrm{rad}_t</math>, which are multiplicative functions which act on prime powers as
<math display=block>\mathrm{rad}_t(p^e) = p^{\mathrm{min}(e, t - 1)}</math>
The cases <math>t=3</math> and <math>t=4</math> are tabulated in and .
The notion of the radical occurs in the abc conjecture, which states that, for any <math>\varepsilon > 0</math>, there exists a finite <math>K_\varepsilon</math> such that, for all triples of coprime positive integers <math>a</math>, <math>b</math>, and <math>c</math> satisfying <math>a+b=c</math>,